A  TREATISE 


ASTRONOMY, 

SPHERICAL  AND   PHYSICAL; 


VTITH 


ASTRONOMICAL   PROBLEMS, 


AND 


SOLAR,  LUNAR,  AND  OTHER  ASTRONOMICAL  TABLES. 


FOR  THE   USE  OF 


COLLEGES   AND    SCIENTIFIC    SCHOOLS. 


BT 


WILLIAM  A.  NORTON,  M.A., 

PROFESSOR  OP  CIVIL   ENGINEERING    IN    TALE    COLLEGE. 


FOURTH  EDITION. 
REVISED,  REMODELLED,  AND  ENLARGED. 


NEW  YORK: 

JOHN  WILEY  &  SON,  PUBLISHERS, 

2  CLINTON  HALL,  ASTOR  PLACE. 

1872. 


a? 


Entered  according  to  Act  of  Congress,  in  the  year  1867, 

BY  WILLIAM  A.  NORTON, 
la  the  Clerks  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District  of  New  York. 


PREFACE  TO  THE  REVISED  EDITION. 


IN  the  preparation  of  the  present  edition  the  work  has  been 
entirely  remodelled.  The  chapters  which  treat  of  Astronomical 
Instruments,  Comets,  the  Fixed  Stars,  and  the  Tides,  and  the 
portion  of  the  chapter  on  the  Sun,  that  treats  of  the  Sun's 
Spots  and  Physical  Constitution,  and  the  Zodiacal  Light,  have 
been  wholly,  or  mostly,  rewritten.  Several  changes  of  plan 
and  arrangement  have  been  made  with  the  view  of  facilitating 
study  and  class  instruction.  The  more  difficult  investigations 
of  astronomical  formulae,  occurring  in  the  text  of  the  former 
editions,  have  been  transferred  to  the  Appendix.  On  the  other 
hand,  the  text  has  been  enlarged  by  giving  a  more  extended  de- 
scription of  astronomical  facts  and  appearances,  and  a  more  com- 
plete discussion  of  physical  phenomena,  including  a  detail  of 
the  important  results  of  recent  investigations  concerning  the 
physical  constitution  of  the  different  classes  of  heavenly  bodies, 
and  a  succinct  exposition  of  the  physical  theories  that  have  been 
generally  received,  or  explain  the  phenomena  most  satisfactorily. 
Such  theoretical  discussions  are  kept  distinct  from  the  universally 
recognized  truths  of  the  science.  The  results  of  the  author's 
own  investigations  on  the  physical  constitution  and  phenomena 
of  Comets,  and  on  the  physical  constitution  of  the  Sun,  and 
the  origin  of  the  Sun's  Spots,  are  briefly  given  in  the  same  con- 
nection. "New  theoretical  views  are  offered,  in  a  note  in  the 
Appendix,  on  the  possible  development  of  sidereal  systems  from 
primordial  nebulous  masses ;  under  the  operation  of  recognized 
material  forces,  originated  and  sustained  by  the  Creator,  which 


303568 


IV  PREFACE. 

unceasingly  execute  His  will.  Some  prominence  is  given  to  tlie 
author's  theory  of  the  variable  intensity  of  the  repulsive  force  of 
the  Sun,  acting  on  different  portions  of  cometic  matter,  as  the 
operative  cause  of  the  lateral  dispersion  of  the  nebulous  matter 
that  makes  up  the  train  of  a  comet.  This  is  believed  to  have 
been  substantiated  by  a  detailed  discussion  and  comparison  with 
observations ;  and  as  recent  astronomical  treatises,  published  in 
this  country  and  in  Europe,  have  advocated  it  without  making 
mention  of  its  previous  publication  and  mathematical  discussion, 
it  is  but  just  and  proper  that  it  should  be  distinctly  set  forth  in 
the  present  work. 

The  Astronomical  Problems  in  Part  III.  remain  substantially 
the  same  as  in  the  last  edition.  The  table  of  Latitudes  and 
Longitudes  of  Places,  the  tables  of  the  Planetary  Elements,  and 
the  table  of  the  Mean  Places  of  Fixed  Stars,  have  been  replaced 
by  others  that  are  more  accurate  and  more  extended.  The  tables 
of  the  Sun's  and  Moon's  Epochs  have  been  extended  to  1884. 
Many  new  illustrative  figures,  and  several  plates  of  telescopic 
appearances,  have  been  added. 

One  of  the  most  important  of  the  special  improvements  intro- 
duced consists  in  the  adoption  of  the  new  and  more  accurate 
determination  of  the  Sun's  parallax,  and  mean  distance  from 
the  earth.  This  is  now  generally  adopted,  as  one  evidence  of 
which  may  be  mentioned  its  introduction  into  the  computations 
of  the  English  Nautical  Almanac  for  1870.  It  brings  with  it 
a  more  accurate  determination  of  the  distances  of  all  the  planets 
from  the  Sun,  and  of  the  satellites  from  their  primaries,  and  the 
dimensions  and  densities  of  the  planets.  The  present  is  the  first 
American  treatise  in  which  this  important  advance  in  exact 
astronomical  science  has  been  incorporated. 

Another  improvement  is  the  insertion  of  a  brief  description 
of  the  methods  used  in  the  United  States  Coast  Survey  in  deter- 
mining from  astronomical  observations  the  latitude  and  longitude 
of  a  place.  These  may  be  characterized  as  the  American 


PREFACE.  V 

methods,  as  they  were  devised  and  perfected  by  American  as- 
tronomers and  engineers;  and  are  superior  to  all  others  that 
have  yet  been  tried. 

Without  further  specification  of  alterations  and  supposed 
improvements,  it  is  hoped  that  the  work  will  be  found,  in 
all  its  features,  a  true  exposition,  within  the  limits  necessarily 
prescribed,  of  the  present  condition  of  the  sublime  science  of 
Astronomy;  from  both  the  theoretical  and  practical  point  of 
view. 

A  large  number  of  astronomical  treatises  and  scientific  peri- 
odicals have  been  consulted.  Professor  Chauvenet's  admirable 
work  on  Spherical  and  Practical  Astronomy,  should  be  particu- 
larly mentioned  as  having  been  especially  consulted  in  prepar- 
ing the  chapter  on  Instruments.  In  the  mention  of  new  dis- 
coveries and  theoretical  views,  as  well  as  of  the  signal  advances 
v^hich  modern  Astronomy  has  made,  the  name  of  the  discoverer, 
or  author,  is  generally  given.  The  history  of  Astronomy  can- 
not properly  be  wholly  omitted  from  a  text-book  on  the  science, 
although  it  may  be  simpler  to  present  the  science  as  a  body  of 
admitted  truths,  without  making  mention  of  their  discovery. 
The  author  takes  occasion  here  to  acknowledge  his  obligations 
to  Professor  C.  S.  Lyman,  of  Yale  College,  for  important  advice, 
and  valuable  assistance  frequently  rendered. 


TABLE  OF  CONTENTS. 


PAET  I. 

SPHERICAL  ASTRONOMY. 

CHAPTER  L 

PAOl 

Definitions  and  Fundamental  Conceptions. — General  Phenomena  of 
the  Heavens .        •  1 

CHAPTER  IL 
Celestial  and  Terrestial  Spheres 11 

CHAPTER  m. 

Astronomical  Instruments. — Astronomical  Observation   ...  24 

The  Transit  Instrument 30 

Astronomical  Clock 

Meridian  Circle 

Altitude  and  Azimuth  Instrument 43 

Equatorial 45 

Sextant 

Errors  of  Instrumental  Admeasurement 51 

The  Telescope 52 

CHAPTER  IV. 

Corrections  of  Measured  Angles •       •  54 

Refraction 54 

Parallax 60 

Aberration        ....• 66 

CHAPTER  Y. 

ifigure  and  Dimensions  of  the  Earth. — Latitude  and  Longitude  of 
a  Place  .  72 

Determination  of  the  Latitude  and  Longitude  of  a  Place  .        .         75 


Viii  CONTENTS. 

CHAPTER  YI. 

PAOB 

Apparent  Motion  of  the  Sun  in  the  Heavens    .        .       '...       .        .  81 
CHAPTER  VII. 

Precession  of  the  Equinoxes. — Nutation  .        .    ;>  ^        .        » .      , 

Nutation    .        »        .        .        .       '4     r  .'»•"»        •   ••   -^» 

CHAPTER  YIII. 

Measurement  of  Time       .        .        .       Y       .        ....  91 

Different  kinds  of  Time ib. 

Conversion  of  One  Species  of  Time  into  Another     ...  92 
Determination  of  Time,  and  Regulation  of  Clocks  by  Astrono- 
mical Observations       •  ,        >••.    •-,    .. .>••:    •'+        ...  94 
The  Calendar     .    ".     (.'.        .  ' 98 

CHAPTER  IX. 

Motions  of  the  Sun,  Moon  and  Planets  in  their  Orbits     .        .        .  102 

Kepler's  Laws    ..-...'  .-:  ..  ^  \        .        .        .        .        .        .  ib. 

Definitions  of  Terms 106 

Elements  of  the  Orbit  of  a  Planet 108 

Determination  of  the  Sun's  Apparent  Orbit,  or  the  Earth's 

Real  Orbit      .     v .     -  ./ - 109 

Mean  Motion     .        .        ^ ib> 

Semi-Major  Axis       »       ~»    ;    •        .        .        .        .        .        .  ib. 

Eccentricity 110 

Longitude  and  Epoch  of  the  Perigee 112 

Determination  of  the  Elements  of  the  Moon's  Orbit         .        .  113 

Longitude  of  the  Node  •  .     ' '. "   "  .     '  .  :/ '  .' '  ^  .        .        .  ib. 

Inclination  of  the  Orbit     .     '  .     -:  .      .  * ,  \.        ,        .        .  114 

Mean  Motion     .     *  .     •  .     v.        •        .        .        .        .        .  ib. 

Longitude  of  the  Perigee,  Eccentricity,  and  Semi-Major  Axis  .  115 

Mean  Longitude  at  an  Assigned  Epoch      .        .        .        .        .  116 

Determination  of  the  Elements  of  a  Planet's  Orbit  .        .        .  ib. 

Heliocentric  Longitude  of  the  Ascending  Node         .        .        .  117 

Inclination  of  the  Orbit     .        .        .        .        .        .        .        .  118 

Periodic  Time    .        ,       -7       ,.       > 119 

To  find  the  Heliocentric  Longitude   and  Latitude,  and  the 

Radius  Yector,  for  a  given  time     ......  ib. 

Longitude  of  the  Perihelion,  Eccentricity,  and  Semi-Major  Axis  121 

Epoch  of  the  Perihelion  Passage        .         .        .        .        .        .  122 

True  and  Mean  Elements .  123 

CHAPTER  X. 

Determination  of  the  Place  of  a  Planet,  or  of  the  Sun  or  Moon,  for 
a  given  time,  by  the  Elliptic  Theory. — Verification  of  Kepler's 

Laws  126 


CONTENTS.  IX 

turn 

Place  01  a  Planet  in  its  Orbit 126 

Heliocentric  Place  of  a  Planet 127 

Geocentric  Place  of  a  Planet 128 

Places  of  the  Bun  and  Moon t'fc. 

Verification  of  Kepler's  Laws 129 

CHAPTER  XI. 

Inequalities  of  the  Motions  of  the  Planets  and  of  the  Moon. — Tables 

for  Finding  the  Places  of  these  Bodies 130 

Tables  of  the  Sun,  Moon,  and  Planets 135 

CHAPTER  XII. 

Motions  of  the  Comets 136 

Halley's  Comet HO 

Encke's  Comet 141 

Biela's  Comet 142 

Fayes'  Comet t&. 

Lexell's  Comet  of  1770      ........  143 

The  Great  Comet  of  1843 144 

Donati's  Comet 145 

Conspicuous  Comets  of  the  Present  Century     ....  146 

CHAPTER  XIII. 

Motions  of  the  Satellites 147 

CHAPTER  XIV. 

The  Sun  and  the  Phenomena  Attending  its  Appafent  Motions         .  151 

Inequality  of  Days *&• 

Twilight 156 

The  Seasons 159 

Form  and  Dimensions  of  the  Sun 162 

Sun's  Spots,  and  Rotation  on  its  Axis. — Physical  Constitution 

of  the  Sun 164 

Zodiacal  Light 175 

CHAPTER  XV. 

The  Moon  and  its  Phenomena 180 

Phases  of  the  Moon *6. 

Moon's  Rising,  Setting,  and  Passage  Over  the  Meridian    .        .  182 

Rotation  and  Librations  of  the  Moon        .....  184 

Dimensions  and  Physical  Constitution  of  the  Moon  .        .        .  186 

Description  of  the  Moon's  Surface     ......  188 

CHAPTER  XVI. 

Eclipses  cf  the  Sun  and  Moon. — Occultations  of  the  Fixed  Stars     .  190 


X  CONTENTS. 

FAGB 

Eclipses  of  the  Moon         .        .        .        .        .'"x?-;»        .  190 

Calculation  of  an  Eclipse  of  the  Moon       .        .        .                •  194 

Construction  of  an  Eclipse  of  the  Moon 198 

Eclipses  of  the  Sun   ......        ^    •  '•>:       .  199 

Calculation  of  an  Eclipse  of  the  Sun         .        .        .      -%:.;•  206 

Occultations       .        .        .        .        .   •     *     '   •        .        .        •.  *&• 

CHAPTER  XVII. 

The  Planets  and  the  Phenomena  Occasioned  by  their  Motions  in 

Space 208 

Apparent  Motions  of  the  Planets  with  Respect  to  the  Sun       .  ib. 

Phases  of  the  Inferior  Planets 214 

Transits  of  the  Inferior  Planets          .        .        .        .        .        .  215 

Appearance,  Dimensions,  Rotation,  and  Physical  Constitution 

of  the  Planets 216 

Mercury     .        .        .        .        ,        . 217 

Yenus        .        .        .-".'.' 218 

Mars.        .         .         .        /:;'  ^       ..        .  V-  .-'%.    '   .,      -.,  220 

Jupiter  and  its  Satellites    .        .        .        „        ..    "   . ;       .        .  221 

Saturn,  with  its  Satellites  and  Ring  .        .        .'        .        .        .  223 

Uranus  and  its  Satellites    .        .        .        ...<'..  -^  ...        .        .  227 

Neptune    .      '  ,  •    .'  *        .        .        '.        .    '    .        «  ib. 

The  Planetoids  .        .  XY.        .     ^  .  ^     .,  228 

CHAPTER  XYIII. 

Comets     .        .        .     ;  ,V. 229 

Their  General  Appearance. — Varieties  of  Appearance       .        .  ib. 

Form,  Structure  and  Dimensions  of  Comets      ....  233 

Physical  Constitution  of  Comets       .....        .        .  236 

Constitution  and  Mode  of  Formation  of  Tails  of  Comets  .        .  237 

Condition  and  Origin  of  the  Nebulous  Envelopes     .        .        .  241 

CHAPTER  XIX. 

The  Fixed  Stars .,;      ...      .  .    ..  *  245 

Constellations. — Division  into  Magnitudes         ....  ib. 

Number  and  Distribution  over  the  Heavens      ....  247 

Annual  Parallax,  and  Distance  of  the  Stars       .       ' .        .        .  250 

Nature  and  Magnitude  of  the  Stars  .        .        .        .        .        .  252 

Variable  Stars    .        .        .      •;        .'      .        .        .        .        „  253 

Double  Stars      .        .:     .        , 256 

Proper  Motions  of  the  Stars      ;        .^     . .        .        .        .        .  259 

Clusters  of  Stars        ,        «  '     v'1  '".        .        .      ..        .        .  261 

Nebulae      .        .        .        ...     «•  *   V     '.  +  r      »                .  ib. 

Distance  and  Magnitude  of  Nebulae 267 

Number,  Mutual  Distance,  and  Comparative  Brightness  of  the 

Component  Stars  of  Clusters                                           .  269 


CONTENTS.  XI 

*AO« 

Structure  of  the  Sidereal  Universe 270 

General  Dynamical  Condition  of  Sidereal  Systems   .        .        .  273 

CHAPTER  XX. 

Theories  of  the  Evolution  of  Sidereal  and  Planetary  Systems          .  275 

Nebular  Hypothesis ib. 

Development  of  the  Solar  System ib. 


PAET  II. 

PHYSICAL  ASTRONOMY. 

CHAPTER  XXI. 
Principle  of  Universal  Gravitation 278 

CHAPTER  XXII. 

Theory  of  the  Elliptic  Motion  of  the  Planets 282 

CHAPTER  XXIII. 

Theory  of  the  Perturbations  of  the  Elliptic  Motions  of  the  Planets 
and  the  Moon 287 

CHAPTER  XXIV. 

Relative  Masses  and  Densities  of  the  Sun,  Moon,  and  Planets. — 
Relative  Intensity  of  the  Force  of  Gravity  at  their  Surface .        .        297 

CHAPTER  XXV. 

Form  and  Density  of  the  Earth. — Changes  of  its  Period  of  Rota- 
tion.— Precession  of  the  Equinoxes,  and  Nutation        .        .        .        299 

CHAPTER  XXVI. 

The  Tides 302 

Comparison  of  the  Theory  of  the  Tides  with  the  Results  of 

Observation 307 

Tides  of  the  Atlantic  Coast  of  the  United  States      ...  309 

Tides  of  the  Pacific  Coast 311 

Tides  of  the  Gulf  of  Mexico 312 

Tides  of  the  Mediterranean       .        .        .                .        .        .  313 

Tides  of  Inland  Seas  and  Lakes t&. 

Tides  of  the  Coast  of  Europe ib. 

Establishment  of  the  Port. — Tide  Tables  .  & 


Xii  CONTENTS. 

(  PAET  HI. 

ASTRONOMICAL  PROBLEMS. 

PAOB 

EXPLANATIONS  OF  THE  TABLES    .        .        .     -  .        •,'-''•      "*  •>'  •        31^ 

PROB.  I.  To  work  by  logistical  logarithms  a  proportion,  the  terms 
of  which  are  degrees  and  minutes,  or  minutes  and  seconds,  of 
arc ;  or  hours  and  minutes,  or  minutes  and  seconds,  of  time  .  320 

PROB.  II.  To  take  from  a  table  the  quantity  corresponding  to  a 
given  value  of  the  argument,  or  to  given  values  of  the  argu- 
ments of  the  table  ....  .  •••  •-;*  .  •  •  321 

PROB.  III.  To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equa- 
tor into  Hours,  Minutes,  &c.,  of  Time  .;  ••;•".  .  .  .  327 

PROB.  IV.     To  convert  Time  into  Degrees,  Minutes,  and  Seconds   .          ib. 

PROB.  V.     The  Longitudes  of  two  Places,  and  the  Time  at  one  of 

them  being  given,  to  find  the  corresponding  time  at  the  other        328 

PROB.  VI.  The  Apparent  Time  being  given,  to  find  the  correspond- 
ing Mean  Time ;  or,  the  Mean  Time  being  given,  to  find  the 
Apparent  .  .  -'•,-••  •*-,'  *«  ,  •*  'V  >  •  •  329 

PROB.  VII.     To  correct  the  Observed  Altitude  of  a  Heavenly  Body 

for  Refraction    .      '.,.•.;«..';•    .'      *        *,      •        .        •        •        332 

PROB.  VIII.     The  Apparent  Altitude  of  a  Heavenly  Body  being 

given,  to  find  its  True  Altitude .        j    '     •        .        .        .        •        333 

PROB.  IX.  To  find  the  Sun's  Longitude,  Hourly  Motion,  and  Semi- 
diameter,  for  a  given  Time,  from  the  Tables  ....  335 

PROB.  X.     To  find  the  Apparent  Obliquity  of  the  Ecliptic,  for  a 

given  time,  from  the  Tables       .        .        ,        . .       .        .        .        337 

PROB.  XI.     Given  the  Sun's  Longitude  and  the  Obliquity  of  the 

Ecliptic,  to  find  his  Right  Ascension  and  Declination        .        .        338 

PROB.  XII.     Given  the  Sun's  Right  Ascension  and  the  Obliquity  of 

the  Ecliptic,  to  find  his  Longitude  and  Declination   .        .        .        339 

PROB.  XIII.     The  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic 

being  given,  to  find  the  Angle  of  Position        ....          ib. 

PROB.  XIV.  To  find  from  the  Tables,  the  Moon's  Longitude,  Lati- 
tude, Equatorial  Parallax,  Semi-diameter,  and  Hourly  Motions 
in  Longitude  and  Latitude,  for  a  given  Time  ....  340 

PROB.  XV.     The  Moon's  Equatorial  Parallax,  and  the  Latitude  of 

a  Place,  being  given,  to  find  the  Reduced  Parallax  and  Latitude        349 

PROB.  XVI.     To  find  the  Longitude  and  Altitude  of  the  Nonagesi- 

mal  Degree  of  the  Ecliptic,  for  a  given  Time  and  Place    .        .          ib. 

PROB.  XVII.  To  find  the  Apparent  Longitude  and  Latitude,  as 
affected  by  Parallax,  and  the  Augmented  Semi-diameter  of  the 
Moon ;  the  Moon's  True  Longitude,  Latitude,  Horizontal  Semi- 
diameter,  and  Equatorial  Parallax,  and  the  Longitude  and  Alti- 
tude of  the  Nonagesimal  Degree  of  the  Ecliptic,  being  given  .  352 


CONTENTS. 


PROB.  XVIII.    To  find  the  Mean  Right  Ascension  and  Declination, 

or  Longitude  and  Latitude  of  a  Star,  for  a  given  Time,  from  the 

Tables        ...........        356 

PROB.  XIX.     To  find  the  Aberrations  of  a  Star  in  Right  Ascension 

and^Declination  for  a  given  Day        ......        357 

PROB.  XX.     To  find  the  Nutations  of  a  Star  in  Right  Ascension 

and  Declination,  for  a  given  Day       .        .        .        fc        .        .        358 
PROB.  XXI.     To  find  the  Apparent  Right  Ascension  and  Declina- 

tion of  a  Star,  for  a  given  Day  .......        360 

PROB.  XXII.     To  find  the  Aberrations  of  a  Star  in  Longitude  and 

Latitude,  for  a  given  Day  .  ......        361 

PROB.  XXIII.     To  find  the  Apparent  Longitude  and  Latitude  of  a 

Star,  for  a  given  Day         ........          ib. 

PROB.  XXIV.    To  Compute  the  Longitude  and  Latitude  of  a  Heav- 

enly Body  from  its  Right  Ascension    and  Declination,   the 

Obliquity  of  the  Ecliptic  being  given         .....        362 
PROB.  XXV.     To  compute  the  Right  Ascension  and  Declination 

of  a  Heavenly  Body  from  its  Longitude  and  Latitude,   the 

Obliquity  of  the  Ecliptic  being  given        .....        363 
PROB.  XXVI.     The  Longitude  and  Declination  of  a  Body  being 

given,  and  also  the  Obliquity  of  the  Ecliptic,  to  find  the  Angle 

of  Position        .......        ...        364 

PROB.  XXVII.     To  find  from  the  Tables  the  Time  of  New  or  Full 

Moon,  for  a  given  Year  and  Month  ......        365 

PROB.  XXVIII.     To  determine  the  number  of  Eclipses  of  the  Sun 

and  Moon  that  may  be  expected  to  occur  in  any  given  Year, 
.  and  the  Times  nearly  at  which  they  will  take  place  .        .        .        368 
PROB.  XXIX.    To  calculate  an  Eclipse  of  the  Moon        .        .        .        371 
PROB.  XXX.    To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place        375 
PROB.  XXXI.    To  find  the  Moon's  Longitude,  &c.,  from  the  Nau- 

tical Almanac    ,  392 


APPENDIX. 

TRIGONOMETRICAL  FORMULA 395 

I.  Relative  to  a  Single  Arc  or  angle  a ib. 

II.  Relative  to  Two  Arcs  a  and  b,  of  which  a  is  supposed  to  be 

the  greater ib. 

III.  Trigonometrical  Series 397 

IV.  Differences  of  Trigonometrical  Lines  .....  ib. 

V.  Resolution  of  Right-angled  Spherical  Triangles    .        .        .  ib. 

VI.  Resolution  of  Oblique-angled  Spherical  Triangles       .        .  399 


CONTENTS. 

MOB 

INVESTIGATION  OP  ASTRONOMICAL  FORMULJB 402 

Formulae  for  the  Parallax  in  Eight  Ascension  and  Declination, 

and  in  Longitude  and  Latitude ib. 

Formulae  for  the  Aberration  in  Longitude  and  Latitude,  and  in 

Right  Ascension  and  Declination  ....*.        409 
Formulae  for  the  Nutation  in  Right  Ascension  and  Declination        413 
Formulae  for  computing  the  effects  of  the  Oblateness  of  the 
Earth's  Surface,  upon  the  Apparent  Zenith  Distance   and 

Azimuth  of  a  Star 417 

Solution  of  Kepler's  Problem,  by  which  a  Body's  Place  is  found 

in  an  Elliptical  Orbit 418 

Formulae  for  calculating  the  Parallax  in  Altitude  of  a  Heavenly 

Body,  from  its  True  Zenith  Distance 421 

Formulae  for  computing  the  Annual  Variations  in  the  Right 

Ascension  and  Declination  of  a  Heavenly  Body    .        .        .        422 
Formulae  for  computing  the  Heliocentric  Longitude  and  Latitude 
and  Radius  Vector  of  a  Planet,  from  its  Geocentric  Longitude 

and  Latitude  .       <. 423 

Formulae  for  computing  the  Geocentric  Longitude  and  Latitude 
of  a  Planet,  from  its  Heliocentric  Longitude  and  Latitude  and 

Radius  Vector .        .        .        424 

Calculation  of  an  Eclipse  of  the  Sun .        •    ''.•_       .        .        .        426 
Calculation  of  an  Occultation     .        .        .        .        .        .        .        431 

NOTE  I.    Construction  of  Tables      .        .        .    '    .        .        .        .        432 

NOTE  II.    Relative  to  Sun's  Spots   .'..•-•     •        *        •        .        434 

NOTE  III.    Kirk-wood's  Law    .        .      '* 436 

NOTE  IV.    Relative  to  Origin  of  Comets  ..        ....        437 

NOTEV.    Origin  of  Sidereal  Systems 439 


ASTRONOMY. 


PAET  I. 

SPHERICAL    ASTROISTOMY. 


CHAPTER   I. 

DEFINITIONS  AND  FUNDAMENTAL  CONCEPTIONS;  GENERAL 
PHENOMENA  OF  THE  HEAVENS. 

1.  The  sun,  moon,  and  stars — the  luminous  bodies  dissemi- 
nated through  the  heavens,  or  indefinite  space  surrounding  the 
earth — are  called  Heavenly  Bodies.     The  heavenly  bodies,  consi- 
dered collectively,  are  often  termed  the  Heavens.     The  science 
which  treats  of  the  heavenly  bodies  is  called  Astronomy.     It  is 
divided  into  Theoretical  and  Practical  Astronomy.     Theoretical 
Astronomy  is  divided  into  Spherical  and  Physical  Astronomy. 

2.  Spherical  Astronomy  treats  of  the  positions,  motions,  and 
distances  of  the  heavenly  bodies ;  and  of  their  appearance,  mag- 
nitude, form,  and   structure.     It  comprises  the  theory  of  the 
methods  of  observation  and  calculation  by  which  the  positions, 
motions,  etc.,  of  the  heavenly  bodies  have  been  determined ;  and 
the  whole  body  of  exact  knowledge  thus  acquired,  which  is  often 
termed  Descriptive  Astronomy. 

Physical  Astronomy  investigates  the  general  physical  cause  of 
the  motions  and  constitution  of  the  bodies  of  the  material  uni- 
verse, and  deduces  from  this  general  cause,  called  the  force  of 
universal  gravitation,  all  the  details  of  the  celestial  mechanism. 

Practical  Astronomy  treats  of  astronomical  instruments,  and 
astronomical  observation ;  practical  determinations,  as  of  the 
latitude  or  longitude  of  a  place,  from  instrumental  observation  ; 
and  the  solution  of  astronomical  problems  with  the  aid  of 
tables. 


2  FUNDAMENTAL  CONCEPTIONS. 

3.  Form  of  tlie  Earth.  We  learn  from  the  following  cir- 
cumstances that  the  earth  is  a  body  of  a  globular  form,  insulated 
in  space. 

(1.)  When  a  vessel  is  receding  from  the  land,  an  observer, 
from  a  point  on  the  coast,  first  loses  sight  of  the  hull,  then  of 
the  lower  parts  of  the  sails,  and  lastly  of  the  topsails.  It  will  be 
readily  perceived,  on  glancing  at  Fig.  1,  that  no  part  of  the  earth 
could  become  interposed  between  the  hull,  and  then  the  lower  por- 
tions of  the  sails  of  a  distant  vessel,  and  the  eye  of  the  observer, 
if  the  sea  were  really  what  it  appears  to  be,  an  indefinitely 
extended  plane ;  also  that  if  the  earth  be  round,  a  receding  ship 
should  disappear  in  the  manner  it  is  actually  observed  to  do,  as 
the  hull,  mainsail,  and  topsails  pass  in  succession  below  the  line 
of  sight  tangent  to  the  surface  of  the  sea.  If  the  observer  take 
a  more  elevated  position  the  ship  should  begin  to  sink  out  of 
sight  at  a  greater  distance,  because  the  line  of  sight  will  touch 
the  sea  at  a  more  distant  point. 


FIG.  i. 


(2.)  At  sea  the  visible  horizon,  or  the  line  bounding  the  visible 
portion  of  the  earth's  surface,  is  everywhere  a  circle,  of  a  greater 
or  less  extent  according  to  the  altitude  of  the  point  of  observa- 
tion, and  is  on  all  sides  equally  depressed.  To  illustrate  this 
proof,  let  BOA  (Fig.  2)  represent  a  portion  of  the  earth's  sur- 


face supposed  to  be  spherical,  P  the  position  of  the  eye  of  the 
observer,  and  DPC  a  horizontal  line.     If  we  conceive  lines,  such 


VISIBLE   PORTION  OF  THE   HEAVENS.  3 

as  PA  and  PB,  to  be  drawn  through  the  point  of  observation  P, 
tangent  to  the  earth  in  every  direction,  it  is  plain  that  these  lines 
will  all  touch  the  earth  at  the  same  distance  from  the  observer, 
and  therefore  that  the  line  AGB,  conceived  to  be  traced  through 
all  the  points  of  contact,  A,  B,  etc.,  which  would  be  the  visible 
horizon,  is  a  circle.  It  is  also  manifest  that  the  angles  of  depres- 
sion CPA,  DPB,  etc.,  of  the  horizon  in  different  directions,  will 
be  equal ;  and  that  a  greater  portion  of  the  earth's  surface  will 
be  seen,  and  thus  that  the  horizon  will  increase  in  extent,  in 
proportion  as  the  altitude  of  the  point  of  observation,  P, 
increases. 

(3.)  Navigators,  as  it  is  well  known,  have  sailed  entirely 
around  the  earth. 

These  facts  prove  the  surface  of  the  sea  to  be  convex,  and 
the  surface  of  the  land  conforms  very  nearly  to  that  of  the  sea ; 
for  the  elevations  of  the  highest  mountains  bear  an  exceedingly 
small  proportion  to  the  dimensions  of  the  whole  earth. 

4-  Visible  and  Invisible  Portions  of  tlie  Heavens.  If 
an  indefinite  number  of  lines,  PA,  PB,  etc.,  be  conceived  to  be 
drawn  through  the  point  of  observation  P,  (Fig.  2,)  touching  the 
earth  on  all  sides,  a  conical  surface  will  be  formed,  having  its 
vertex  at  P,  and  extending  indefinitely  into  space.  All  heavenly 
bodies,  which  at  any  time  are  situated  below  this  surface,  have 
the  earth  interposed  between  them  and  the  eye  of  the  observer, 
and  therefore  cannot  be  seen.  All  bodies  that  are  above  this 
surface,  which  send  sufficient  light  to  the  eye,  are  visible.  That 
portion  of  the  heavens  which  is  above  this  surface,  presents  the 
appearance  of  a  solid  vault  or  canopy,  resting  upon  the  earth  at 
the  visible  horizon,  (see  Fig.  2 ;)  and  to  this  vault  the  sun,  moon, 
and  stars  seem  to  be  attached.  It  is  hardly  necessary  to  remark 
that  this  is  an  optical  illusion.  It  will  be  seen  in  the  sequel  that 
the  heavenly  bodies  are  distributed  through  space  at  various  dis- 
tances from  the  earth,  and  that  the  distances  of  all  of  them  are 
very  great  in  comparison  with  the  dimensions  of  the  earth. 

It  will  suffice,  in  the  conception  of  phenomena,  to  suppose  the 
eye  of  the  observer  to  be  near  the  earth's  surface,  and  that  the 
imaginary  conical  surface  above  mentioned,  which  separates  the" 
visible  from  the  invisible  portion  of  the  heavens,  is  a  horizontal 
plane,  confounded  for  a  certain  distance  with  the  visible  part  of 
the  earth.  This  is  called  the  plane  of  the  horizon,  and  sometimes 
the  horizon  simply. 

5.  Up  and  down,  at  any  place  on  the  earth's  surface,  are  from 
and  towards  the  surface ;  and  thus  at  different  places  have  every 
variety  of  absolute  direction  in  space. 

6.  The  Sky.     The  earth  is  surrounded  with  a  transparent 
gaseous  medium,  called  the  earth's  atmosphere,  estimated  to  be 
some  fifty  miles  in  height;    through  which  all  the  heavenly 
bodies  are  seen.     The  atmosphere  is  not  perfectly  transparent, 


4  GENERAL  PHENOMENA. 

but  shines  throughout  with  light  received  from  the  heavenly 
bodies,  and  reflected  from  its  particles;  and  thus  forms  a  lumi- 
nous canopy  over  our  heads  by  day  and  by  night.  This  ia 
called  the  Sky.  It  appears  blue  because  this  is  the  color  of  the 
atmosphere;  that  is,  because  the  particles  of  the  atmosphere 
reflect  the  blue  rays  more  abundantly  than  any  other  color. 
By  day  the  portion  of  the  atmosphere  which  lies  above  the 
horizon  is  highly  illuminated  by  the  sun,  and  shines  with  so 
strong  a  light  as  to  efface  the  stars. 

7.  Diurnal  Motion  of  the  Heavens.  The  most  conspi- 
cuous of  the  celestial  phenomena,  is  a  continual  motion  common 
to  all  the  heavenly  bodies,  by  which  they  are  carried  around  the 
earth  in  regular  succession.  The  daily  circulation  of  the  sun  and 
moon  about  the  earth  is  a  fact  recognised  by  all  persons.  If  the 
heavens  be  attentively  watched  on  any  clear  evening,  it  will 
soon  be  seen  that  the  stars  have  a  motion  precisely  similar  to 
that  of  the  sun  and  moon.  To  describe  the  phenomenon  in 
detail,  as  witnessed  at  night: — if,  on  a  clear  night,  we  observe 
the  heavens,  we  shall  find  that  the  stars,  while  they  retain  the 
same  situations  with  respect  to  each  other,  undergo  a  continual 
change  of  position  with  respect  to  the  earth.  Some  will  be 
seen  to  ascend  from  a  quarter  called  the  East,  being  replaced 
by  others  that  come  into  view,  or  rise;  others,  to  descend  towards 
the  opposite  quarter,  the  West,  and  to  go  out  of  view,  or  set:  and 
if  our  observations  be  continued  throughout  the  night,  with  the 
east  on  our  left,  and  the  west  on  our  right,  the  stars  which  rise  in 
the  east  will  be  seen  to  move  in  parallel  circles,  entirely  across  the 
visible  heavens,  and  finally  to  set  in  the  west.  Each  star  will 
ascend  in  the  heavens  during  the  first  half  of  its  course,  and  de- 
scend during  the  remaining  half.  The  greatest  heights  of  the 
several  stars  will  be  different,  but  they  will  all  be  attained  towards 
that  part  of  the  heavens  which  lies  directly  in  front,  called  the 
South.  If  we  now  turn  our  backs  to  the  south,  and  direct  our 
attention  to  the  opposite  quarter,  the  North,  new  phenomena  will 
present  themselves.  Some  stars  will  appear,  as  before,  ascending, 
reaching  their  greatest  heights,  and  descending ;  but  other  stars 
will  be  seen,  further  to  the  north,  that  never  set,  and  which  appear 
to  revolve  in  circles,  from  east  to  west,  about  a  certain  star  that 
seems  to  remain  stationary.  This  seemingly  stationary  star  is 
called  the  Pole  Star ;  and  the  stars  which  revolve  about  it,  and 
never  set,  are  called  Circumpolar  Stars.  It  should  be  remarked, 
however,  that  the  pole  star,  when  accurately  observed  by  means 
of  instruments,  is  found  not  to  be  strictly  stationary,  but  to  de- 
scribe a  small  circle  about  a  point  at  a  little  distance  from  it  as  a 
fixed  centre.  This  point  is  called  the  North  Pole.  It  is,  in  reality, 
about  the  north  pole,  as  thus  defined,  and  not  the  pole  star,  that 
the  apparent  revolutions  of  the  stars  at  the  north  are  performed. 
At  the  corresponding  hours  of  the  following  night  the  aspect  of 


THE  PLANETS.  5 

the  heavens  will  be  the  same,  from  which  it  appears  that  the  stars 
return  to  the  same  position  once  in  about  24  hours.  It  would 
seem,  then,  that  the  stars  all  appear  to  move  from  east  to  west 
exactly  as  if  attached  to  the  concave  surface  of  a  hollow  sphere, 
which  rotates  in  this  direction  about  an  axis  passing  through  the 
station  of  the  observer  and  the  north  pole  of  the  heavens,  in  a 
space  of  time  nearly  equal  to  24  hours.  For  the  sake  of  simpli- 
city this  conception  is  generally  adopted.  This  motion,  common 
to  all  the  heavenly  bodies,  is  called  their  Diurnal  Motion.  It  is 
ascertained,  by  certain  accurate  methods  of  observation  and  com- 
putation, that  the  diurnal  motion  of  the  stars  is  strictly  uniform 
and  circular. 

§.  Rotating  Sphere  of  the  Heavens.  It  is  important  to 
observe,  that  the  conception  of  a  single  sphere  to  which  the  stars 
are  supposed  to  be  attached,  will  not  represent  their  diurnal 
motion,  as  seen  from  every  part  of  the  earth's  surface,  unless  the 
sphere  be  supposed  to  be  of  such  vast  dimensions  that  the  earth 
is  comparatively  but  a  mere  point  at  its  centre. 

A  circle  cut  out  of  the  heavens  conceived  to  be  a  rotating 
sphere,  by  a  plane  passing  through  the  axis  of  rotation,  has  a  north 
and  south  direction. 

9.  Fixed  Stars  and  Planets.     The  greater  number  of  the 
stars  constantly  preserve  the  same  relative  positions,  and  are 
therefore  called  Fixed  Stars.     But  there  are  also  many  stars 
which  are   perpetually  changing  their  places  in  the  heavens. 
These  are  called  Planets,  or  wandering  stars.     Each  planet  has 
received  a  distinctive  name.     For  convenience  of  designation 
they  are  divided  into  the  two  classes  of  Planets,  and  Planetoids 
or   Minor  Planets.      The  former  class  comprises   the  planets 
Mercury,  Yenus,  Mars,  Jupiter,  Saturn,  Uranus,  and  Neptune. 
The  first  five  of  these  are  visible  to  the  naked  eye ;  but  Uranus 
and  Neptune,  and  the  planetoids,  cannot  be  seen  without  the  aid 
of  a  telescope  ;  and  have  all  been  discovered  since  the  year  1780. 
Table  II.  (a),  p.  5,  &c.  contains  a  list  of  the  planetoids  at  present 
known,  with  the  date  and  place  of  discovery  of  each,  and  the 
name  of  the  discoverer.     The  number  of  planetoids  hitherto 
discovered  is  ninety-one.     Every  year  adds  one  or  more  to  the 
list. 

10.  Distinctive  Peculiarities  of  Different  Planets.  The 
planets  are  distinguishable  from  each  other,  either  by  a  difference 
of  aspect,  or  by  a  difference  of  apparent  motion  with  respect  to 
the  sun.     Venus  and  Jupiter  are  the  two  most  brilliant  planets. 
They  are  quite  similar  in  appearance,  but  their  apparent  motions 
with  respect  to  the  Sun  are  very  different.     Thus  Yenus  never 
recedes  beyond  40°  or  50°  from  the  Sun,  while  Jupiter  is  seen  at 
every  variety  of  angular  distance  from  him.     Mars  is  known  by 
the  ruddy  color  of  his  light.     Saturn  has  a  pale,  dull  aspect. 

11.  Apparent  Motions  of  the  Planets.     The  apparent  mo- 


6 


GENERAL  PHENOMENA. 


tion  of  each  of  the  planets,  is  generally  directed  towards  the  east; 
but  they  are  occasionally  seen  moving  towards  the  west.  As 
their  easterly  prevails  over  their  westerly  motion,  they  all,  in 
process  of  time,  accomplish  a  revolution  around  the  earth.  The 
periods  of  revolution  are  different  for  each  planet. 

12.  Apparent  Motions  of  the  Snii  and  Moon.  The  sun 
and  moon,  are  also  continually  changing  their  places  among 
the  fixed  stars.  From  repeated  observations  of  its  position 
among  the  stars,  it  is  found  that  the  moon  has  a  progressive 
circular  motion  in  the  heavens  from  west  to  east,  and  completes  a 
revolution  around  the  earth  in  about  27  days. 

The  motion  of  the  sun,  is  also  constantly  progressive,  and 
directed  from  west  to  east.  This  will  appear  on  observing  for  a 
number  of  successive  evenings,  the  stars  which  first  become 
visible  in  that  part  of  the  heavens  where  the  sun  sets.  It  will 
be  found  that  the  stars,  which  in  the  first  instance  were  observed 
to  set  just  after  the  sun,  soon  cease  to  be  visible,  and  are  replaced 
by  others  that  were  seen  immediately  to  the  east  of  them ;  and 
that  these  in  their  turn,  give  place  to  others  situated  still  further 
to  the  east.  The  sun  must  then  be  continually  approaching  the 
stars  that  lie  on  the  eastern  side  of  him.  To  make  this  morq 
evident,  let  us  suppose  that  the  small  circle  aon  (Fig.  3)  repre- 


Fig.  3. 


sents  a  section  of  the  earth  perpendicular  to  the  axis  of  rotation 
of  the  imaginary  sphere  of  the  heavens,  (8,*)  conceived  to  pass 
through  the  earth's  centre ;  the  large  circle  H  Z  S  a  section  of 

*  .Numbers  thus  inclosed  in  a  parenthesis  refer  to  a  previous  article. 


APPAKENT  MOTION   OF  THE  SUN.  7 

the  heavens  perpendicular  to  the  same  line,  and  passing  through 
the  sun ;  and  the  right  line  H  o  r  the  plane  of  the  horizon  at  the 
station  o.  The  direction  of  the  diurnal  motion  is  from  H  towards 
Z  and  S.  Suppose  that  an  hour  or  so  after  sunset  the  sun  is  at 
S,  and  that  the  star  r  is  seen  in  the  western  horizon ;  also  that 
the  stars  t,  u,  v,  &c.,  are  so  distributed  that  the  distances  rt,  tu,  uv, 
&c.,  are  each  equal  to  S  r.  Then,  at  the  end  of  two  or  three 
weeks,  an  hour  after  sunset  the  star  t  will  be  in  the  horizon ;  at 
the  end  of  another  interval  of  two  or  three  weeks  the  star  u  will 
be  in  the  same  situation  at  the  same  hour ;  at  the  end  of  another 
interval,  the  star  v,  &c.  It  is  plain,  then,  that  the  sun  must  at 
the  ends  of  these  successive  intervals  be  in  the  successive  posi- 
tions in  the  heavens,  r,  t,  u,  &c. ;  otherwise,  when  it  is  brought 
by  its  diurnal  motion  to  the  point  S,  below  the  horizon,  the 
stars  t,  u,  v,  &c.,  could  not  be  successively  in  the  plane  of  the 
horizon  at  r.  Whence  it  appears  that  the  sun  has  a  motion  in 
the  heavens  in  the  direction  S  r  t  u  v,  opposite  to  that  of  the  di- 
urnal motion  ;  that  is,  towards  the  east. 

Another  proof  of  the  progressive  motion  of  the  sun  among  the 
stars  from  west  to  east,  is  found  in  the  fact  that  the  same  stars 
rise  and  set  earlier  each  successive  night,  and  week,  and  month 
during  the  year.  At  the  end  of  six  months  the  same  stars  rise* 
and  set  12  hours  earlier ;  which  shows  that  the  sun  accomplishes 
half  a  revolution  in  this  interval.  At  the  end  of  a  year,  or  of 
365  days,  the  stars  rise  and  set  again  at  the  same  hours,  from 
which  it  appears  that  the  sun  completes  an  entire  revolution  in 
the  heavens  in  this  period  of  time. 

It  is  to  be  observed  that  the  sun  does  not  advance  directly 
towards  the  east.  It  has  also  some  motion  from  south  to  north, 
and  north  to  south.  It  is  a  matter  of  common  observation  that 
the  sun  is  moving  towards  the  north  from  winter  to  summer,  and 
towards  the  south  from  summer  to  winter. 

When  the  place  of  the  sun  in  the  heavens  is  accurately  found 
from  day  to  day  by  certain  methods  of  observation,  hereafter  to 
be  explained,  it  appears  that  his  path  is  an  exact  circle,  inclined 
about  23°  to  a  circle  running  due  east  and  west  (8). 

1£.  The  Zodiac.  The  motions  of  the  sun,  moon,  and  pla- 
nets, are  for  the  most  part  confined  to  a  certain  zone,  of  about  18° 
in  breadth,  extending  around  the  heavens  obliquely  from  west 
to  east,  which  has  received  the  name  of  the  Zodiac. 

14.  Comet*.  There  is  yet  another  class  of  bodies,  called 
Comets,  or  hairy  Stars,  that  have  a  motion  among  the  fixed 
stars.  They  appear  only  occasionally  in  the  heavens,  and 
continue  visible  only  for  a  few  weeks  or  months.  They 
shine  with  a  diffusive  nebulous  light,  and  are  commonly 
accompanied  by  a  fainter  divergent  stream  of  similar  light, 
called  a  tail. 

The  motions  of  the  comets  are  not  restricted  to  the  zodiac. 


8  GENERAL  PHENOMENA. 

These  bodies  are  seen  in  all  parts  of  the  heavens,  and  moving  in 
every  variety  of  direction. 

15.  Satellites.     By  inspecting  the  planets  with  telescopes,  it 
has  been  discovered  that  some  of  them  are  constantly  attended 
by  a  greater  or  less  number  of  small  stars,  whose  positions  are 
continually  varying.     These  attendant  stars  are  called  Satellites. 
The  planets  which  have  satellites  are  Jupiter,  Saturn,  Uranus, 
and  Neptune.     The  satellites  are  sometimes  called  Secondary 
Planets ;  the  planets  upon  which  they  attend  being  denominated 
Primary  Planets. 

16.  The  Solar  System.     The  sun  and  moon,  the  planets, 
(including  the  earth,)  together  with   their  satellites,  and   the 
comets,  compose  the  Solar  System. 

From  the  consideration  of  the  apparent  motions  and  other 
phenomena  of  the  solar  system,  several  theories  have  been  form- 
ed in  relation  to  the  arrangement  and  actual  motions  in  space  of 
the  bodies  that  compose  it.  The  theory,  or  system,  now  univer- 
sally received,  is,  in  its  most  prominent  features,  that  which  was 
taught  by  Copernicus  in  the  sixteenth  century,  and  which  is 
known  by  the  name  of  the  Copernican  System.  It  is  as  follows : 

The  sun  occupies  a  fixed  centre,  about  which  the  planets  (in- 
cluding the  earth)  revolve  from  west  to  east,  in  planes  that  are 
but  slightly  inclined  to  each  other,  and  in  the  following  order : 
Mercury,  Venus,  the  Earth,  Mars,  the  Planetoids,  Jupiter,  Sa- 
turn, Uranus,  and  Neptune.  The  earth  rotates  from  west  to 
east,  about  an  axis  inclined  to  the  plane  of  its  orbit  about  66^°, 
and  which  remains  continually  parallel  to  itself  as  the  earth 
revolves  around  the  sun.  The  moon  revolves  from  west  to  east 
around  the  earth  as  a  centre ;  and  in  like  manner  the  satellites 
circulate  from  west  to  east  around  their  primaries.  Without  the 
solar  system,  and  at  immense  distances  from  it  are  the  fixed  stars. 

A  motion  in  space  from  west  to  east,  is  a  motion  from  right  to 
left,  as  observed  from  a  station  within  the  orbit  described,  and  on 
the  north  side  of  its  plane.  To  obtain  a  clear  conception  of  the 
motions  of  the  planets,  the  reader  should  place  himself  in  ima- 
gination at  or  near  the  centre  of  the  system,  and  on  the  north 
side  of  the  plane  of  the  earth's  orbit  within  which  the  planets 
may  all,  for  the  present,  be  conceived  to  revolve. 

17.  Symbols.     The  principal  planets,  and  the  sun  and  moon, 
are  often  designated  by  the  following  conventional  characters  or 
symbols. 

The  Sun,    ....  0  Jupiter,     ....  if 

Mercury,     .     .     .     .  $  Saturn,      .     .     .     .  *> 

Venus,        .     .-V    i  $  Uranus,     .     .     .     .  # 

The  Earth,      ...  0  Neptune,    .    .     .     .  £ 

Mars, $  The  Moon,      .     .     .  J 

18.  Inferior,  and  Superior  Planets.     The  two  planets, 


EFFECTS  OF  THE  EARTH'S  ROTATION.  9 

Mercury  and  Yen  us,  whose  orbits  lie  within  the  earth's  orbit, 
are  called  Inferior  Planets.  The  others  are  called  Superior 
Planets.  The  terms  inferior  and  superior  as  here  used,  have 
merely  the  signification  of  lower  and  higher  in  place,  or  in  posi- 
tion with  respect  to  the  sun,  as  compared  with  the  earth. 

19.  Vast  Distance  of  the  Fixed  Stars.     The  angular  dis- 
tance between  any  two  fixed  stars,  is  found  to  be  the  same  from 
whatever  point  of  the  earth's  surface  it  is-  measured.     It  follows, 
therefore,  that  the  diameter  of  the   earth  is  insensible,  when 
compared  with  the  distance  of  the  fixed  stars;  and  that  with 
respect  to  the  region  of  space  which  separates  us  from  those 
bodies,  the  whole  earth  is  a  mere  point.     Moreover,  the  angular 
distance  between  any  two  fixed  stars,  is  the  same  at  whatever 
period  of  the  year  it  is  measured.     Hence,  if  the  earth  revolves 
around  the  sun,  its  entire  orbit  must  be  insensible  in  comparison 
with  the  distance  of  the  stars. 

20.  Explanation  of  the  Diurnal  Motion  of  the  Hea- 
vens.    On  the  hypothesis  of  the  earth's  rotation,  the  diurnal 
motion  of  the  heavens  is  a  mere  illusion  occasioned  by  the  rota- 
tion of  the  earth.     To  explain  this,  suppose  the  axis  of  the  earth 
to  be  prolonged  till  it  intersects  the  heavens  considered  as  con- 
centric with  the  earth.     Conceive  a  great  circle  to  be  traced 
through  the  two  points  of  intersection,  and  the  point  directly 
overhead,  and  let  the  position  of  the  stars  be  referred  to  this 
circle.     It  will  be  readily  perceived  that  the  relative  motion  of 
this  circle  and  the  stars  will  be  the  same,  whether  the  circle 
rotates  with  the  earth  from  west  to  east,  or,  the  earth  being  sta- 
tionary, the  whole  heavens  rotate  about  the  same  axis  and  at  the 
same  rate  in  the  opposite  direction.     Now,  as  the  motion  of  the 
earth  is  perfectly  equable,  we  are  insensible  of  it,  and  therefore 
attribute  the  changes  in  the  situations  of  the  stars  with  respect 
to  the  earth  to  an  actual  motion  of  these  bodies.     It  follows, 
then,  that  we  must  conceive  the  heavens  to  rotate  as  above  men- 
tioned, since,  as  we  have  seen,  such  a  motion  would  give  rise  to 
the  same  changes  of  situation  as  the  supposed  rotation  of  the 
earth.     It  was  stated  (7)  that  the  sphere  of  the  heavens  appears 
to  rotate  about  a  line  passing  through  the  north  pole  and  the 
station  of  the  observer;  but,  as  the  radius  of  the  earth  is  insensi- 
ble in  comparison  with  the  distance  of  the  stars,  an  axis  passing 
through  the  centre  of  the  earth  will  apparently  pass    through 
the  station  of  the  observer,  wherever  this  may  be  upon  the  earth's 
surface. 

21.  Explanation  of  the  Sun's  apparent  Motion.      *W*e 
in  like  manner  infer  that  the  observed  motion  of  the  sun  in  the 
heavens  is  only  an  apparent  motion,  occasioned  by  the  orbital 
motion  of  the  earth.     Let  E,  E'  (Fig.  4)  represent  two  positions 
of  the  earth  in  its  orbit  EE'E"  about  the  sun  S.     When  the 
earth  i<*  at  E,  the  observer  will  refer  the  sun  to  that  part  of  the 


10 


GENERAL  PHENOMENA. 


heavens  marked  5;  but  when  the  earth  is  arrived  at  E',  he  will 
refer  it  to  the  part  marked  s' ;  and  being  in  the  mean  time  in- 
sensible of  his  own  motion,  the  sun  will  appear  to  him  to  have 
described  in  the  heavens  the  arc  s  sf,  just  the  same  as  if  it  had 


.  4. 


actually  passed  over  the  arc  SS'  in  space,  and  the  earth  had, 
during  that  time,  remained  quiescent  at  E.  The  motion  of  the 
sun  from  s  towards  s'  will  be  from  west  to  east,  since  the  motion 
of  the  earth  from  E  towards  E'  is  in  this  direction.  Moreover, 
as  the  axis  of  the  earth  is  inclined  to  the  plane  of  its  orbit  under 
an  angle  of  66£°  (16),  the  plane  of  the  sun's  apparent  path, 
which  is  the  same  as  that  of  the  earth's  orbit,  will  be  inclined 
23£°  to  a  circle  perpendicular  to  the  earth's  axis,  or  to  a  circle 
directed  due  east  and  west. 


CELESTIAL  AND  TERRESTIAL  SPHERES.  11 


CHAPTER  H. 

CELESTIAL  AND  TERRESTRIAL  SPHERES. 

22.  Celestial  Sphere.  In  determining  from  observation  the 
apparent  positions  and  motions  of  the  heavenly  bodies,  and,  in 
general,  in  all  investigations  that  have  relation  to  their  apparent 
positions  and  motions,  astronomers  conceive  all  these  bodies, 
whatever  may  be  their  actual  distance  from  the  earth,  to  be 
referred  to  a  spherical  surface  of  an  indefinitely  great  radius,  hav- 
ing the  station  of  the  observer,  or  what  comes  to  the  very  same 
thing,  the  centre  of  the  earth,  for  its  centre.  This  imaginary  spheri- 
cal surface  is  called  the  Sphere  of  Hie  Heavens,  or  the  Celestial  Sphere. 
It  is  important  to  observe,  that  by  reason  of  the  great  dimensions 
of  this  sphere,  if  two  lines  be  drawn  through  any  two  points  of 
the  earth,  and  parallel  to  each  other,  they  will,  when  indefinitely 
prolonged,  meet  it  sensibly  in  the  same  point ;  and  that,  if  two 
parallel  planes  be  passed  through  any  two  points  of  the  earth, 
they  will  intersect  it  sensibly  in  the  same  great  circle.  This 
amounts  to  saying  that  the  earth,  as  compared  to  this  sphere,  is  to  be 
considered  as  a  mere  point  at  its  centre. 

Not  only  is  the  size  of  the  earth  to  be  neglected  in  comparison 
with  the  celestial  sphere,  but  also  the  size  of  the  earth's  orbit. 
Thus  the  supposed  annual  motion  of  the  earth  around  the  sun, 
does  not  change  the  point  in  which  a  line  conceived  to  pass  from 
any  station  upon  the  earth  in  any  fixed  direction  into  space, 
pierces  the  sphere  of  the  heavens ;  nor  the  circle'  in  which  a 
plane  cuts  the  same  sphere. 

The  fixed  stars  are  so  remote  from  the  earth  that  observers, 
wherever  situated  upon  the  earth,  and  in  the  different  positions 
of  the  earth  in  its  orbit,  refer  them  to  the  same  points  of  the  celes- 
tial sphere,  (19.)  The  other  heavenly  bodies  are  referred  by 
observers  at  different  stations  to  points  somewhat  different. 

Definitions.  For  the  purposes  of  observation  and  compu- 
tation, certain  imaginary  points,  lines,  and  circles,  appertaining 
to  the  celestial  sphere,  are  employed,  which  we  shall  now  proceed 
to  define  and  explain. 

(1.)  The  Vertical  Line,  at  any  place  on  the  earth's  surface,  is 
the  line  of  descent  of  a  falling  body,  or  the  position  assumed  by 
a  plumb-line  when  the  plummet  is  freely  suspended  and  at  rest. 

Every  plane  that  passes  through  the  vertical  line  is  called  a 


12 


CELESTIAL  SPHERE. 


Vertical  Plane.     Every  plane  that  is  perpendicular  to  the  vertical 
line,  is  called  a  Horizontal  Plane. 

(2.)  The  Sensible  Horizon  of  a  place  on  the  earth's  surface,  is 
the  circle  in  which  a  horizontal  plane  drawn  through  the  place, 
cuts  the  celestial  sphere.  As  its  plane  is  tangent  to  the  earth,  it 
separates  the  visible  from  the  invisible  portion  of  the  heavens, 

(4.) 

(8.)  The  Rational  Horizon  is  a  circle  parallel  to  the  former,  the 
plane  of  which  passes  through  the  centre  of  the  earth.  The  zone 
of  the  heavens  comprehended  between  the  sensible  and  rational 
horizon  is  imperceptible,  or  the  two  circles  appear  as  one  and  the 
same  at  the  distance  of  the  earth. 

(4.)  The  Zenith  of  a  place  is  the  point  in  which  the  vertical 
line  prolonged  upwards  pierces  the  celestial  sphere*.  The  point 
in  which  the  vertical  line,  when  produced  downwards,  intersects 
the  celestial  sphere,  is  called  the  Nadir. 

The  zenith  and  nadir  are  the  geometrical  poles  of  the  horizon. 

(5.)  The  Axis  of  the  Heavens  is  an  imaginary  right  line  passing 
through  the  north  pole  (7)  and  the  centre  of  the  earth.  It  is 
the  line  about  which  the  apparent  rotation  of  the  heavens  is  per- 
formed. It  is,  also,  on  the  hypothesis  of  the  earth's  rotation,  the 
axis  of  rotation  of  the  earth  prolonged  on  to  the  heavens. 

(6.)  The  South  Pole  of  the  heavens  is  the  point  in  which  the 
axis  of  the  heavens  meets  the  southern  part  of  the  celestial 
sphere. 


To  illustrate  the  preceding  definitions,  let  the  inner  circle  nOs 

Fig.  5)  represent  the  earth,  and  the  outer  circle   HZRN    the 

sphere  of  the  heavens ;  also  let  0  be  a  point  on  the  earth's  sur- 


CELESTIAL   SPHERE. 


13 


face,  and  OZ  the  vertical  line  at  the  station  0.— Then  HOR 
will  be  the  plane  of  the  sensible  horizon,  HCR  the  plane  of  the 
rational  horizon,  Z  the  zenith,  and  N  the  nadir ;  and  if  P  be  the 
north  pole  of  the  heavens,  OP,  or  CP  its  parallel,  will  be  the 
axis  of  the  heavens,  and  P'  the  south  pole. 

The  lines  CP  and  OP  intersect  the  heavens  in  the  same  point, 
P ;  and  the  planes  HOR,  and  HCR,  in  the  same  circle,  passing 
through  the  points  H  and  R. 

Unless  we  are  seeking  for  the  exact  apparent  place  in  the  hea- 
vens of  some  other  heavenly  body  than  a  fixed  star,  we  may  con- 
ceive the  observer  to  be  stationed  at  the  earth's  centre,  in  which 
case  OP  will  become  the  same  as 
CP,  and  HOR  the  same  as  HCR; 
as  represented  in  Fig.  6.  In  this 
diagram,  the  circle  of  the  horizon 
being  supposed  to  be  viewed  from 
a  point  above  its  plane,  is  repre- 
sented by  the  ellipse  HARa,  Z 
and  N  are  its  geometrical  poles. 
In  the  construction  of  Fig.  5,  the 
eye  is  supposed  to  be  in  the  plane 
of  the  horizon,  and  HARa  is  pro- 
jected into  its  diameter  HCR. 

Every  different  place  on  the 
surface  of  the  earth  has  a  different 
zenith,  and  except  in  the  case  of  diametrically  opposite  places,  a 


FIG.  7. 


different  horizon.     To  illustrate  this,  let  nesq  (Fig.  7)  represent 
the  earth,  and  HZRP'  the  sphere  of  the  heavens;  then  considering 


14  CELESTIAL  SPHERE. 

the  four  stations,  e,  O,  n,  and  q,  the  zenith  and  horizon  of  the  first 
will  be  respectively  E  and  PeP' ;  of  the  second  Z  and  HOE  ;  of 
the  third  P  and  QnE  ;  of  the  fourth  Q  and  P'^P.  The  diametri- 
cally opposite  places  0  and  0'  have  the  same  rational  horizon, 
viz.  HOE.  The  same  is  true  of  the  places  n  and  5,  and  e  and  q. 
Their  rational  horizons  are  respectively  QCE  and  PGP'. 

(7.)  Vertical  Circles  are  great  circles  passing  through  the  zenith 
and  nadir.  They  cut  the  horizon  at  right  angles,  and  their 
planes  are  vertical.  Thus  ZSM  (Fig.  6)  represents  a  vertical 
circle  passing  through  the  star  S,  called  the  Vertical  Circle  of  the 
Star. 

(8,)  The  Meridian  of  a  place  is  that  vertical  circle  which  con- 
tains the  north  and  south  poles  of  the  heavens.  The  plane  of  the 
meridian  is  called  the  Meridian  Plane. 

Thus,  PZEP'  is  the  meridian  of  the  station  C.  The  half 
HZE,  above  the  horizon,  is  termed  the  Superior  Meridian,  and 
the  other  half  ENH,  below  the  horizon,  is  termed  the  Inferior 
Meridian.  The  two  points,  as  H  and  E,  in  which  the  meridian 
cuts  the  horizon,  are  called  the  North  and -South  Points  of  the 
horizon;  and  the  line  of  intersection,  as  HCE,  of  the  meridian 
plane  with  the  plane  of  the  horizon,  is  called  the  Meridian  Line, 
or  the  North  and  South  Line. 

(9.)  The  Prime  Vertical  is  the  vertical  circle  which  crosses  the 
meridian  at  right  angles.  It  cuts  the  horizon  in  two  points,  as 
e,  w,  called  the  East  and  West  Points  of  the  Horizon. 

(10.)  The  Altitude  of  any  heavenly  body  is  the  arc  of  a  vertical 
circle,  intercepted  between  the  centre  of  the  body  and  the  horizon, 
or  the  angle  at  the  centre  of  the  sphere,  measured  by  this  arc. 
Thus,  SM  or  MCS  is  the  altitude  of  the  star  S. 

(11.)  The  Zenith  Distance  of  a  heavenly  body  is  the  arc  of  a 
vertical  circle,  intercepted  between  its  centre  and  the  zenith  ;  or 
the  distance  of  the  centre  of  the  body  from  the  zenith  as  mea- 
isured  by  the  arc  of  a  great  circle.  Thus,  ZS,  or  ZCS,  is  the 
.senith  distance  of  the  star  S. 

It  is  obvious  that  the  zenith  distance  and  altitude  of  a  body  are 
complements  of  each  other,  and  therefore  when  either  one  is  known 
that  the  other  may  be  found. 

(12.)  The  Azimuth  of  a  heavenly  body  is  the  arc  of  the  horizon, 
intercepted  between  the  meridian  and  the  vertical  circle  passing- 
through  the  centre  of  the  body ;  or  the  angle  comprehended  be- 
tween the  meridian  plane  and  the  vertical  plane  containing  the 
centre  of  the  body.  It  is  reckoned  either  from  the  north  or  from 
the  south  point,  and  each  way  from  the  meridian.  HM  or  HCM 
represents  the  azimuth  of  the  star  S. 

The  Azimuth  and  Altitude,  or  azimuth  and  zenith  distance  of 
a  heavenly  body,  ascertain  its  position  with  respect  to  the  horizon 
and  meridian,  and  therefore  its  place  in  the  visible  hemisphere. 
Thus,  the  azimuth  HM  determines  the  position  of  the  vertical  cir- 


DEFINITIONS.  15 

cle  ZSM  of  the  star  S  with  respect  to  the  meridian  ZPH,  and  the 
altitude  MS,  or  the  zenith  distance  ZS,  the  position  of  the  star  in 
this  circle. 

(13.)  The  Amplitude  of  a  heavenly  body  at  its  rising,  is  the  arc 
of  the  horizon  intercepted  between  the  point  where  the  body  rises 
and  the  east  point.  Its  amplitude  at  setting,  is  the  arc  of  the  ho- 
rizon intercepted  between  the  point  where  the  body  sets  and  the 
west  point.  It  is  reckoned  towards  the  north,  or  towards  the 
south,  according  as  the  point  of  rising  or  setting  is  north  or  south 
of  the  east  or  west  point.  Thus,  if  aBSA  represents  the  circle 
described  by  the  star  S  in  its  diurnal  motion,  ea  will  be  its  ampli- 
tude at  rising,  and  wA  its  amplitude  at  setting. 

(14.)  The  Celestial  Equator,  or  the  Equinoctial,  is  a  great  circle 
of  the  celestial  sphere,  the- plane  of  which  is  perpendicular  to  the 
axis  of  the  heavens.  The  north  and  south  poles  of  the  heavens 
are  therefore  its  geometrical  poles.  The  celestial  equator  is 
represented  in  Fig.  6  by  Ez^Qe.  This  circle  is  also  frequently 
called  the  Equator,  simply. 

(15.)  Parallels  of  Declination  are  small  circles  parallel  to  the 
celestial  equator.  aBSA  represents  the  parallel  of  declination 
of  the  star  S. 

The  parallels  of  declination  passing  through  the  stars,  are  the 
circles  described  by  the  stars  in  their  apparent  diurnal  motion. 
These,  by  way  of  abbreviation,  we  shall  call  Diurnal  Circles. 

(16.)  Celestial  Meridians,  Hour  Circles,  and  Declination  Cir- 
cles, are  different  names  given  to  all  great  circles  which  pass 
through  the  poles  of  the  heavens,  cutting  the  equator  at  right 
angles.  PSP' is  a  celestial  meridian.  The  angles  comprehended 
between  the  hour  circles  and  the  meridian,  reckoning  from  the 
meridian  towards  the  west,  are  called  Hour  Angles,  or  Horary 
Angles. 

(17.)  The  Ecliptic  is  that  great  circle  of  the  heavens  which  the 
sun  appears  to  describe  in  the  course  of  the  year. 

(18.)  The  Obliquity  of  the  Ecliptic  is  the  angle  under  which  the 
ecliptic  is  inclined  to  the  equator.  Its  amount  is  23£°. 

(19.)  The  Equinoctial  Points  are  the  two  points  in  which  the 
ecliptic  intersects  the  equator.  That  one  of  these  points  which 
the  sun  passes  in  the  spring  is  called  the  Vernal  Equinox,  and 
the  other,  which  is  passed  in  the  autumn,  is  called  the  Autumnal 
Equinox.  These  terms  are  also  applied  to  the  epochs  when  the 
sun  is  at  the  one  or  the  other  of  these  points.  These  epochs  are, 
for  the  vernal  equinox  the  21st  of  March,  and  for  the  autumnal 
equinox  the  23d  of  September,  or  thereabouts. 

(20.)  The  Solstitial  Points  are  the  two  points  of  the  ecliptic  90° 
distant  from  the  vernal  and  autumnal  equinox.  The  one  that 
lies  to  the  north  of  the  equator  is  called  the  Summer  Solstice,  and 
the  other  the  Winter  Solstice.  The  epochs  of  the  sun's  arrival  at 
these  points  are  also  designated  by  the  same  terms.  The  summer 


16 


CELESTIAL  SPHERE. 


solstice  happens  about  the  21st  of  June,  and  the  winter  solstice 
about  the  22d  of  December. 

(21.)  The  Equinoctial  Colure  is  the  celestial  meridian  passing 
through  the  equinoctial  points ;  and  the  Solstitial  Colure  is  the 
celestial  meridian  passing  through  the  solstitial  points. 

(22.)  The  Polar  Circles  are  parallels  of  declination  at  a  distance 
from  the  poles  equal  to  the  obliquity  of  the  ecliptic.  The  one 
about  the  north  pole  is  called  the  Arctic  Circle;  the  other,  about 
the  south  pole,  is  called  the  Antarctic  Circle. 

The  polar  circles  contain  the  geometrical  poles  of  the  ecliptic. 

(28.)  The  Tropics  are  parallels  of  declination  at  a  distance  from 
the  equator  equal  to  the  obliquity  of  the  ecliptic.  That  which  is 
on  the  north  side  of  the  equator  is  called  the  Tropic  of  Cancer, 
and  the  other  the  Tropic  of  Capricorn. 

The  tropics  touch  the  ecliptic  at  the  solstitial  points. 


FIG.  8. 

Let  C  (Fig.  8)  represent  the  centre  of  the  earth  and  sphere, 
POP'  the  axis  of  the  heavens,  EVQA  the  equator,  WVTA  the 
ecliptic,  and  K,  K',  its  poles.  Then  will  V  be  the  vernal  and  A 
the  autumnal  equinox  ;  W  the  winter,  and  T  the  summer  solstice  ; 
PVP'A  the  equinoctial  colure;  PKWK'T  the  solstitial  colure ; 
the  angle  TCQ,  or  its  measure  the  arc  TQ,  the  obliquity  of  the 
ecliptic  ;  KmU,  K'w'U',  the  polar  circles  ;  and  TwZ,  Wn'Z',  the 
tropics. 


DEFINITIONS.  17 

It  is  important  to  observe  that,  agreeably  to  what  has  been 
stated  (p.  11),  the  directions  of  the  equator  and  ecliptic,  of  the 
equinoctial  points,  and  of  the  other  points  and  circles  just  defined 
and  illustrated,  are  the  same  at  any  station  upon  the  surface  of 
the  earth  as  at  its  centre.  Thus,  the  equator  lies  always  in  the 
plane  passing  through  the  place  of  observation,  wherever  this 
may  be,  and  parallel  to  the  plane  which,  passing  through  the 
earth's  centre,  cuts  the  heavens  in  this  circle.  In  like  manner 
the  ecliptic  lies,  everywhere,  in  a  plane  parallel  to  that  which  is 
conceived  to  pass  through  the  centre  of  the  earth  and  cut  the 
heavens  in  this  circle,  and  so  for  the  other  circles. 

(24.)  The  Zodiac  (13)  extends  about  9°  on  each  side  of  the 
ecliptic. 

(25.)  The  ecliptic  and  zodiac  are  divided  into  twelve  equal 
parts,  called  Signs.  Each  sign  contains  30°.  The  division  com- 
mences at  the  vernal  equinox.  Setting  out  from  this  point,  and 
following  around  from  west  to  east,  the  Signs  of  the  Zodiac,  with 
the  respective  characters  by  which  they  are  designated,  are  as 
follows:  Aries  T,  Taurus  8,  Gemini  n,  Cancer  s,  Leo  £L, 
Virgo  TTJT,  Libra  =^,  Scorpio  fll,  Sagittarius  / ,  Capricornus  13*, 
Aquarius  ^,  Pisces  =£.  The  first  six  are  called  northern  signs, 
being  north  of  the  equinoctial.  The  others  are  called  southern  siyns. 

The  vernal  equinox  corresponds  to  the  first  point  of  Aries, 
and  the  autumnal  equinox  to  the  first  point  of  Libra.  The  sum- 
mer solstice  corresponds  to  the  first  point  of  Cancer,  and  the 
winter  solstice  to  the  first  point  of  Capricornus. 

The  mode  of  reckoning  arcs  on  the  ecliptic  is  by  signs, 
degrees,  minutes,  &c. 

A  motion  in  the  heavens  in  the  order  of  the  signs,  or  from 
west  to  east,  is  called  a  direct  motion,  and  a  motion  contrary  to 
the  order  of  the  signs,  or  from  east  to  west,  is  called  a  retrograde 
motion. 

(26.)  The  Right  Ascension  of  a  heavenly  body  is  the  arc  of  the 
equator  intercepted  between  the  vernal  equinox  and  the  declina- 
tion- circle  which  passes  through  the  centre  of  the  body,  as 
reckoned  from  the  vernal  equinox  towards  the  east.  It  mea- 
sures the  inclination  of  the  declination  circle  of  the  body  to  the 
equinoctial  col  tire.  Thus,  PSR  being  the  declination  circle  of 
the  star  S,  and  V  the  place  of  the  vernal  equinox.  VE  is  the 
right  ascension  of  the  star.  It  is  the  measure  of  the  angle  YPS. 
If  PS'R'  be  the  declination  circle  of  another  star  S',  SPS',  or 
RR',  will  be  their  difference  of  right  ascension. 

(27.)  The  Declination  of  a  heavenly  body  is  the  arc  of  a  circle 
of  declination,  intercepted  between  the  centre  of  the  body  and 
the  equator.  It  therefore  expresses  the  distance  of  the  body  from 
the  equator.  Thus,  RS  is  the  declination  of  the  star  S. 

Declination  is  North  or  South,  according  as  the  body  is  north 
or  south  of  the  equator. 

2 


18  CELESTIAL  SPHERE. 

In  the  use  of  formulae,  a  south  declination  is  regarded  as 
negative. 

The  right  ascension  and  declination  of  a  heavenly  body  are  two 
co-ordinates,  which,  taken  together,  fix  its  position  in  the  sphere  of  the 
heavens:  for  they  make  known  its  situation  with  respect  to  two 
circles,  the  equinoctial  colure,  and  the  equator.  Thus,  VR  and 
US  ascertain  the  position  of  the  star  S  with  respect  to  the  circles 
PVP'A  and  VQAE. 

(28.)  The  Polar  Distance  of  a  heavenly  body  is  the  arc  of  a 
declination  circle,  intercepted  between  the  centre  of  the  body  and 
the  elevated  pole.  The  polar  distance  is  the  complement  of  the 
declination,  and,  therefore,  when  either  is  known  the  other  may 
be  found. 

(29.)  Circles  of  Latitude  are  great  circles  of  the  celestial  sphere, 
which  pass  through  the  poles  of  the  ecliptic,  and  therefore  cut 
this  circle  at  right  angles.  Thus,  KSL  represents  a  part  of  the 
circle  of  latitude  of  the  star  S. 

(30.)  The  Longitude  of  a  heavenly  body  is  the  arc  of  the  eclip- 
tic, intercepted  between  the  vernal  equinox  and  the  circle  of 
latitude  which  passes  through  the  centre  of  the  body,  as  reckoned 
from  the  vernal  equinox  towards  the  east,  or  in  the  order  of  the 
signs.  It  measures  the  inclination  of  the  circle  of  latitude  of  the 
body  to  the  circle  of  latitude  passing  through  the  vernal  equinox. 
Thus,  VL  is  the  longitude  of  the  star  S.  It  is  the  measure  of  the 
angle  VKS. 

(31.)  The  Latitude  of  a  heavenly  body  is  the  arc  of  a  circle  of 
latitude,  intercepted  between  the  centre  of  the  body  and  the 
ecliptic.  It  therefore  expresses  the  distance  of  the  body  from 
the  ecliptic.  Thus,  LS  is  the  latitude  of  the  star  S. 

Latitude  is  North  or  South,  according  as  the  body  is  north  or 
south  of  the  ecliptic. 

In  the  use  of  formulas,  a  south  latitude  is  affected  with  the 
minus  sign. 

The  longitude  and  latitude  of  a  heavenly  body  are  another  set  of 
co-ordinates,  which  serve  to  fix  its  position  in  the  heavens.  They 
ascertain  its  situation  with  respect  to  the  circle  of  latitude  pass- 
ing through  the  vernal  equinox,  and  the  ecliptic.  Thus,  YL  and 
LS  fix  the  position  of  the  star  S,  making  known  its  situation 
with  respect  to  the  circles  KVK'A  and  VTAW. 

(32.)  The  Angle  of  Position  of  a  star  is  the  angle  included  at 
the  star  between  the  circles  of  latitude  and  declination  passing 
through  it.  PSK  is  the  angle  of  position  of  the  star  S. 

(33.)  The  Astronomical  Latitude,  or  the  Latitude  of  a  place, 
is  the  arc  of  the  meridian  intercepted  between  the  zenith  and  the 
celestial  equator.  It  is  North  or  South,  according  as  the  zenith 
is  north  or  south  of  the  equator.  ZE  (Fig.  7)  represents  the 
latitude  of  the  station  0;  QOE  or  QCE  being  the  equator. 

23.  Terrestrial  Sphere.     The  earth's  surface,  considered  as 


DEFINITIONS.  19 

spherical  (which  accurate  admeasurement,  upon  principles  that 
will  be  explained  in  the  sequel,  shows  it  to  be,  very  nearly),  is 
called  the  Terrestrial  Sphere.  The  following  geometrical  con- 
structions appertain  to  the  terrestrial  sphere,  as  it  is  employed 
for  the  purposes  of  astronomy.  It  will  be  observed  that  they 
correspond  to  those  of  the  celestial  sphere  above  described,  and 
are  used  for  similar  objects. 

Definitions.  (1.)  The  North  and  South  Poles  of  the  Earth  are 
the  two  points  in  which  the  axis  of  the  heavens  intersects  the 
terrestrial  sphere.  They  are  also  the  extremities  of  the  earth's 
axis  of  rotation. 

(2.)  The  Terrestrial  Equator  is  the  great  circle  in  which  a  plane 
passing  through  the  centre  of  the  earth,  and  perpendicular  to  the 
axis  of  the  heavens  and  earth,  cuts  the  terrestrial  sphere.  The 
terrestrial  and  the  celestial  equator,  then,  lie  in  the  same  plane. 
The  poles  of  the  earth  are  the  geometrical  poles  of  the  terrestrial 
equator.  The  two  hemispheres  into  which  the  terrestrial  equa- 
tor divides  the  earth  are  called,  respectively,  the  Northern  Hemi- 
sphere and  the  Southern  Hemisphere. 

(3.)  Terrestrial  Meridians  are  great  circles  of  the  terrestrial 
sphere,  passing  through  the  north  and  south  poles  of  the  earth, 
and  cutting  the  equator  at  right  angles.  Every  plane  that 
passes  through  the  axis  of  the  heavens  cuts  the  celestial  sphere 
in  a  celestial  meridian,  and  the  terrestrial  sphere  in  a  terrestrial 
meridian. 

Let  PP'  (Fig.  9)  represent  the  axis  of  the  heavens,  0  the  centre 
of  the  earth,  and  p  and  p'  its  poles.  Then,  elq  will  represent  the 
terrestrial  equator  (ELQ  representing  the  celestial  equator) ;  and 
pepr  and  psp'  terrestrial  meridians  (PEP'  and  PSP'  representing 
celestial  meridians). 

(4.)  The  Reduced  Latitude  of  a  place  on  the  earth's  surface  is 
the  arc  of  a  terrestrial  meridian,  intercepted  between  the  place 
and  the  equator,  or  the  angle  at  the  centre  of  the  earth  measured 
by  this  arc.  Thus,  oe,  or  the  angle  oOe,  is  the  reduced  latitude 
of  the  place  o.  Latitude  is  North  or  South,  according  as  the 
place  is  north  or  south  of  the  equator.  The  reduced  latitude  dif- 
fers somewhat  from  the  astronomical  latitude,  by  reason  of  the 
slight  deviation  of  the  earth  from  a  spherical  form.  Their  differ- 
ence is  called  the  Reduction  of  Latitude. 

(5.)  Parallels  of  Latitude  are  small  circles  of  the  terrestrial 
sphere  parallel  to  the  equator.  Every  point  of  a  parallel  of  lati- 
tude has  the  same  latitude. 

The  parallels  of  latitude  which  correspond  in  situation  with, 
the  polar  circles  and  tropics  in  the  heavens,  have  received 
the  same  appellations  as  these  circles.  (See  defs.  22,  23,  p. 
16.) 

(6.)  The  Longitude  of  a  place  on  the  earth's  surface  is  the 
inclination  of  its  meridian  to  that  of  some  particular  station,  fixed 


20 


TERRESTRIAL   SPHERE. 


upon  as  a  circle  to  reckon  from,  and  called  the  First  Meridian. 
It  is  measured  by  the  arc  of  the  equator,  intercepted  between  the 
first  meridian  and  the  meridian  passing  through  the  place,  and  is 
called  East  or  West,  according  as  the  latter  meridian  is  to  the  east 


JIG.  9. 


or  to  the  west  of  the  first  meridian.  Thus,  if  pqp'  be  supposed 
to  represent  the  first  meridian,  the  angle  spy,  or  the  arc  ql,  will 
be  the  longitude  of  the  place  s. 

Different  nations  have,  for  the  most  part,  adopted  different  first 
meridians.  The  English  use  the  meridian  which  passes  through 
the  Royal  Observatory  at  Greenwich,  near  London ;  and  the 
French,  the  meridian  of  the  Imperial  Observatory  at  Paris.  In  the 
United  States  the  longitude  is,  for  astronomical  purposes, 
reckoned  from  the  meridian  of  Washington,  or  of  Green- 
wich. 

The  longitude  and  latitude  of  a  place  designate  its  situation  on  the 
earth's  surface.  They  are  precisely  analogous  to  the  right  ascen- 
sion and  declination  of  a  star  in  the  heavens. 

24.  Altitude  of  the  Pole.  The  diagram  (see  Fig.  6)  that 
was  made  use  of  in  Art.  22,  in  illustrating  the  description  of  the 
circles  of  the  celestial  sphere,  represents  the  aspect  of  this 


ALTITUDE   OF  THE  POLE. 


21 


FIG.  10. 


sphere  at  a  place  at  which  the  north  pole  of  the  heavens  is  some- 
where between  the  zenith 
and  horizon.  Such  is  the 
position  of  the  north  pole 
at  all  places  situated  be- 
tween the  equator  and  the 
north  pole  of  the  earth. 
For,  let  0  (Fig.  10)  repre- 
sent a  place  on  the  earth's 
surface.  II OR  the  horizon, 
OZ  the  vertical,  HZR  the  me- 
ridian, and  ZE  the  latitude. 
QOE  will  then  represent  the 
equinoctial,  and  P,  90°  dis- 
tant from  E  and  on  the  su- 
perior meridian,  the  elevated 
pole.  Xow  we  have 

HP  =  ZH  —  ZP  =  90°  —  ZP ;    ZE  =  PE  —  ZP  =  90°  —  ZP. 
Whence,  HP  — ZE. 

Thus,  the  altitude  of  the  pole  is  everywhere  equal  to  the  latitude 
of  the  place.  It  follows,  therefore,  that  in  proceeding  from  the 
equator  to  the  north  pole,  the  altitude  of  the  north  pole  of  the 
heavens  will  gradually  increase  from  0°  to  90°. 

By  inspecting  Fig.  7,  it  will  be  seen  that  this  increase  of  the 
altitude  of  the  pole  in  going  north,  is  owing  to  the  fact  that  in 
following  the  curved  surface  of  the  earth  the  horizon,  which  is 
continually  tangent  to  the  earth,  is  constantly  more  and  more 
depressed  towards  the  north,  while  the  absolute  direction  of  the 
pole  remains  unaltered. 

If  the  spectator  is  in  the  southern  hemisphere,  the  elevated 
pole,  as  it  is  always  on  the  opposite  side  of  the  zenith  from  the 
equator,  will  be  the  south  pole.  At  corresponding  situations 
of  the  spectator,  it  will  obviously  have  the  same  altitude  as  the 
north  pole  in  the  northern  hemisphere. 

25.  Oblique  Sphere.  Let  us  now  inquire  minutely  into  the 
principal  circumstances  of  the  diurnal  motion  of  the  stars,  as  it  is 
seen  by  a  spectator  situated  somewhere  between  the  equator  and 
the  north  pole.  And  in  the  first  place,  it  is  a  simple  corollary 
from  the  proposition  just  established,  that  the  parallel  of  declina- 
tion to  the  north,  whose  polar  distance  is  equal  to  the  latitude  of 
the  place,  will  lie  entirelv  above  the  horizon,  and  just  touch  it  at 
the  north  point.  This  circle  is  called  the  circle  of  perpetual  appa- 
rition ;  the  line  aH  (Fig.  11)  represents  its  projection  on  the 
meridian  plane.  The  stars  comprehended  between  it  and  the 
north  pole  will  never  set.  As  the  depression  of  the  south  pole  is 
equal  to  the  altitude  of  the  north  pole,  the  parallel  of  declination 
oR,  at  a  distance  from  the  south  pole  equal  to  the  latitude  of  the 


22 


CELESTIAL   SPHERE. 


place,  will  lie  entirely  below  the  horizon,  and  just  touch  it  at 
the  south  point.     The  parallel  thus  situated  is  called  the  rirck 

of  perpetual  occultation.  The 
stars  comprehended  between  it 
and  the  south  pole  will  nevei 


rise. 


The  celestial  equator  (which 
passes  through  the  east  and 
west  points)  will  intersect  the 
meridian  at  a  point  E,  whose 
zenith  distance  ZE  is  equal  to 
the  latitude  of  the  place  (def. 
33,  Art.  22),  and  consequently, 
whose  altitude  RE  is  equal  to 
the  co  latitude  of  the  place.  There- 
fore, in  the  situation  of  the  ob- 
server above  supposed,  the 
equator  QOE,  passing  to  the 
south  of  the  zenith,  will,  to- 
gether with  the  diurnal  circles  nr,  st,  etc.,  which  are  all  parallel 
to  it,  be  obliquely  inclined  to  the  horizon,  making  with  it  an  angle 
equal  to  the  co-latitude  of  the  place.  As  the  centres  c,  c',  etc., 
of  the  diurnal  circles  lie  on  the  axis  of  the  heavens,  which  is 
inclined  to  the  horizon,  all  diurnal  circles  situated  between  the 
two  circles  of  perpetual  apparition  and  occultation,  all  and  oR, 
with  the  exception  of  the  equator,  will  be  divided  unequally  by 
the  horizon.  The  greater  parts  of  the  circles  nr,  n'r',  etc.,  to  the 
north  of  the  equator,  will  be  above  the  horizon  ;  -and  the  greater 
parts  of  the  circles  st,  sY,  etc.,  to  the  south  of  the  equator,  will 
be  below  the  horizon.  Therefore,  while  the  stars  situated  in  the 
equator  will  remain  an  equal  length  of  time  above  and  below 
the  horizon,  those  to  the  north  of  the  equator  will  remain  a  longer 
time  above  the  horizon  than  below  it ;  and  those  to  the  south  of  the 
equator,  on  the  contrary,  a  longer  time  below  the  horizon  than 
above  it.  It  is  also  obvious,  from  the  manner  in  which  the  hori- 
zon cuts  the  different  diurnal  circles,  that  the  disparity  between 
the  intervals  of  time  that  a  star  remains  above  and  below  the 
horizon  will  be  the  greater  the  more  distant  it  is  from  the  equa- 
tor. Again,  the  stars  will  all  culminate,  or  attain  to  their  greatest 
altitude,  in  the  meridian:  for,  since  the  meridian  crosses  the 
diurnal  circles  at  right  angles,  they  will  have  the  least  zenith 
distance  when  in  this  circle.  Moreover,  as  the  meridian  bisects 
the  portions  of  the  diurnal  circles  which  lie  above  the  horizon, 
the  stars  will  all  employ  the  same  length  of  time  in  passing  from 
the  eastern  horizon  to  the  meridian,  as  in  passing  from  the  meri- 
dian to  the  western  horizon.  The  circumpolar  stars  will  pass  the 
meridian  twice  in  24  hours;  once  above,  and  once  below 
the  pole.  These  meridian  passages  are  called,  respectively, 


ASPECTS  OF  THE   CELESTIAL  SPHERE. 


23 


Upper  and  Lower  Culminations,  or  Inferior  and  Superior 
Transits. 

It  will  be  observed,  that  in  travelling  towards  the  north  the 
circles  of  perpetual  apparition  and  occultation,  together  with 
those  portions  of  the  heavens  about  the  poles  which  are  con- 
stantly visible  and  invisible,  are  continually  on  the  increase. 

It  is  evident,  from  what  is  stated  in  Art.  24,  that  the  circum- 
stances of  the  diurnal  motion  will  be  the  same  at  any  place  in  the 
southern  hemisphere,  as  at  the  place  which  has  the  same  latitude 
in  the  northern. 

The  celestial  sphere  in  the  position  relative  to  the  horizon 
which  we  have  now  been  considering,  which  obtains  at  all  places 
situated  between  the  equator  and  either  pole,  is  called  an  Oblique 
Sphere,  because  all  bodies  rise  and  set  obliquely  to  the  horizon. 

26.  Right  Sphere.  When  the  spectator  is  situated  on  the 
equator,  both  the  celestial  poles  will  be  in  his  horizon  (24),  and 
therefore  the  celestial  equator  and  the  diurnal  circles  in  general 
will  be  perpendicular  to  the  horizon.  This  situation  of  the 
sphere  is  called  a  Eight  Sphere,  for  the  reason  that  all  bodies  rise 
and  set  at  right  angles  with  the  horizon.  It  is  represented  in 
Fig.  12.  As  the  diurnal  circles  are  bisected  by  the  horizon, 
the  stars  will  all  remain  the  same  length  of  time  above  as  below 
the  horizon. 


Fia.  12. 


FIG.  13. 


2T.  Parallel  Sphere.  If  the  observer  be  at  either  of  the 
poles,  the  elevated  pole  of  the  heavens  will  be  in  his  zenith  (24), 
and  consequently  the  celestial  equator  will  be  in  his  horizon. 
The  stars  will  move  in  circles  parallel  to  the  horizon,  and  the 
whole  hemisphere,  on  the  side  of  the  elevated  pole,  will  be  con- 
tinually visible,  while  the  other  hemisphere  will  be  continually 
invisible.  This  is  called  a  Parallel  Sphere.  It  is  represented,  in 
Fig.  13. 


24  ASTRONOMICAL  INSTRUMENTS. 


CHAPTER  III. 

ASTRONOMICAL  INSTRUMENTS. — ASTRONOMICAL  OBSERVATION. 

28.  Astronomical  instruments  are  used  to  measure  arcs  of 
the  celestial  sphere,  or  their  corresponding  angles  at  a  station  on 
the  earth.     They  consist,  essentially,   of  a  refracting  telescope 
turning  upon  an  axis,  and  a  graduated  limb,  or  two  graduated 
limbs  at  right  angles  to  each  other,  to  indicate  the  angle  passed 
over  by  the  telescope.     When  designed  to  measure  angles  in 
the  meridian  plane,  the  axis  of  rotation  is  horizontal,  and  a 
single  vertical  limb  is  used. 

29.  Tlie  Reticle.    At  the  common  focus  of  the  object-glass 
and  eye-glass  of  the  telescope,  is  a  piece  of  apparatus  called  a 
reticle,  the  design  of  which  is  to  furnish  a  definite  line  of  sight. 
In  its  simplest  form  it  consists  of  a  flat  circular  ring,  attached  to 
which  are  two  very  fine  wires,  or  spider  lines,  crossing  each 
other  at  right  angles  in  its  centre  (Fig.  14).     The  line  passing 
through   the  point  of  intersection  of  the  cross  wires  and   the 
centre  of  the  object-glass,  indefinitely  prolonged,  is  the  line  of 
sight,  or  Line  of  Collimation  of  the  telescope. 

The  reticle  can  be  moved  up  or  down,  or  to  the  right  or  left, 
by  adjusting  screws;  and  the  line  of 
collimation  thus  made  perpendicular  to 
the  axis  of  rotation  of  the  telescope. 
These  screws  are  shown  at  aa  and  bb, 
Fig.  14.  They  pass  through  narrow 
slits  in  the  tube  of  the  telescope,  so  that 
they  can  be  turned  from  without,  and 
each  pair  of  screws,  aa  or  lib,  gives  a 
motion  to  the  wire-plate.  The  line  pass- 
ing through  the  centre  of  the  eye-glass 
and  the  centre  of  the  object-glass,  or  the 
optical  axis  of  the  telescope,  is  perpendi- 
cular, or  nearly  so,  to  the  axis  of  rotation.  When  it  is  in  that 
precise  position  and  the  line  of  collimation  accurately  adjusted, 
the  two  lines  will  coincide.  But  it  is  not  important  in  the  use  of 
instruments  that  this  coincidence  should  be  perfect.  It  is  suffi- 
cient if  the  line  of  collimation  is  perpendicular  to  the  axis  of 
rotation. 


MOVABLE    MICROMETER  WIRE. 


25 


Reticle  Tube.  The  reticle  is  placed  in  a  tube,  which  slides 
in  the  lower  end  of  the  principal  tube  of  the  telescope.  The 
eyepiece  is  inserted  in  the  outer  end  of  this  tube;  and  can  be 
pressed  in  or  drawn  out  until  the  wires  of  the  reticle  are  dis- 
tinctly seen.  In  making  an  observation,  the  reticle  tube  with 
the  eye-piece  is  moved  out  or  in  if  necessary,  by  means  of  a 
milled  head  screw  that  works  a  pinion  in  a  rack  connected  with 
the  tube,  until  the  image  of  the  star,  formed  by  the  object-glass, 
falls  upon  the  wires,  when  both  the  wires  and  the  star  will  be 
distinctly  seen.  The  reticle  tube  can  also  be  turned  around 
until  the  wires  have  the  right  direction  in  the  field  of  view.  A 
star  is  known  to  be  on  the  line  of  collimation  when  it  is 
bisected  by  each  of  the  two  cross  wires. 

30.  Improved  Form  of  Bolide.     The  form  of  reticle  just 
described  is  now  attached  only  to  portable  instruments.     That 
which  is  adapted  to  the  larger  instruments  of  an  observatory, 
differs  from  this  in  the  number  of  the  wires,  the  form  of  the 
wire  plate,  and  the  mode  of  attaching  the  plate  to  its  tube  and 
of  adjusting  the  wires.     It  has  several  parallel  and  equidistant 
wires,  crossed  at  right  angles  by  a  single  wire,  or  more  com- 
monly by  two  very  close  parallel  wires  (Fig.  15).     In  meridian 
instruments,   and   those   for    measuring 

altitudes,  the  single  wire,  or  the  equiva- 
lent pair  of  close  parallel  wires,  is  made 
horizontal.  The  middle  wire  of  the  others 
is  brought  into  the  meridian  plane ;  these 
are  called  transit  wires.  The  star  is  made 
to  pass  through  the  field  between  the 
two  horizontal  wires.  The  point  of  the 
middle  transit  wire  that  lies  midway 
between  the  two  horizontal  ones,  corre- 
sponds to  the  point  of  intersection  of  the  two  cross  wires  in 
Fig  H. 

Tlte  luire-plate  lies  within  a  frame  fastened  across  the  outer 
end  of  the  reticle  tube  (SCQ  Fig.  19,  p.  31),  and  is  adjusted  by 
screws  that  act  upon  pieces  projecting  from  its  outer  rim.  The 
eye-piece  is  screwed  into  a  plate  that  slides  within  the  same  trans- 
verse frame,  and  is  moved  by  means  of  a  screw.  By  turning  this 
screw  the  eye-piece  may  be  brought  into  such  positions  that  the 
star  observed  is  kept  in  the  middle  of  the  field  of  view. 

31.  Movable  Micrometer  Wire.     In  the  focus  of  the  eye- 
glass there  is  often  fastened  to  a  transverse  sliding  plate,  and 
movable  with  it,  a  wire  at  right  angles  to  the  direction  in  which 
the  plate  is  moved  by  a  screw.      The  screw  has  the  form  of  the 
micrometer-screw  with   graduated   head,  soon  to  be  described. 
This  wire  is  called  the  movable  micrometer-wire,  and  the  whole 
apparatus,  being  especially  designed  for  the  measurement  of 
small  angular  distances,  is  called  a  Micrometer.     The  same  name 


FIG.  15. 


26  ASTRONOMICAL  INSTRUMENTS. 

is  sometimes,  though  improperly,  given  to  the  reticle  alone,  when 
the  movable  wire  is  not  employed. 

32.  Reading  off  the  Angle.     The  telescope,  and  the  gradu- 
ated limb  which  is  perpendicular  to  the  axis  of  rotation,  are,  in  most 
instruments,  firmly  attached  to  each  other,  and  turn  together  about 
this  axis.     The  lirnb  glides  past  a  fixed  index.     The  angle  read 
off  is  that  which  is  pointed  out  by  the  index.    The  limbs  of  even 
the  largest  instruments,  are  not  divided  into  smaller  parts  than 
2' ;  but  by  means  of  certain  subsidiary  contrivances,  the  angle 
may,  with  some  instruments,  be  read  off  to  within  a  fraction  of  a 
second.     The  principal  contrivances  in  use  for  increasing  the 
accuracy  of  the  reading  off  of  angles,  are  the  Vernier,  and  the 
Reading  Microscope. 

33.  The  Vernier  is  simply  the  index-plate  so  graduated  that 
a  certain  number  of  its  divisions  occupy  the  same  space  as  a 
number  one  less  on  the  limb.     A  division,  or  space  on  the  ver- 
nier, will  therefore  be  less,  by  a  certain  amount,  say  1',  than  a 
division  on  the  limb.     The  index  will,  therefore,  have  moved  I/, 
or  2',  or  3',  etc.,  beyond  the  last  line  of  division  on  the  limb, 
passed  before  it  became  stationary,  according  as  the  first,  second, 
third,  etc.,  line  of  division  of  the  vernier  beyond  the  index  coin- 
cides with  a  line  of  division  on  the  limb. 

In  Fig.  16,  MN  represents  a  portion  of  the  lirnb  of  an  instru- 
ment, divided  into  degrees  and  10'  spaces ;  V  the  Yernier,  tea 


FIG. 


equal  divisions  of  which  have  the  same  extent  as  nine  of  the  10' 
spaces  on  the  limb  ;  and  A  the  index-arm,  which  is  here  sup- 
posed to  revolve  with  the  telescope.  The  index-point,  or  zero  of 
the  vernier,  is  seen  to  be  just  beyond  the  point  30°  10'  on  the 
limb  ;  and  on  looking  along  the  vernier,  we  perceive  that  the 
fourth  line  of  division  from  the  zero  coincides  with  one  of  the 
lines  of  division  of  the  limb.  The  zero  of  the  vernier  is,  there- 
fore, 4'  beyond  the  point  30°  10'  on  the  limb  ;  and  the  whole 
reading  is  30°  14'.  It  is  here  implied  that  one  of  the  divisions 
of  the  vernier  is  less  by  V  than  the  10'  space  on  the  limb.  To 


THE  READING  MICROSCOPE. 


27 


show  this,  let  x  =  the  Dumber  of  minutes  in  a  division  of  the 
vernier,  then  by  what  is  stated  above, 

10  x  =  9  X  10' ;  whence  x  =  9',  and  10'  —  x  =  V. 

By  increasing  the  number  of  divisions  of  the  vernier  that 
corresponds  to  a  number  one  less  on  the  limb,  the  angle  may  be 
read  off  more  accurately.  For  example,  if  sixty  divisions  of  the 
vernier  were  made  equal  to  fifty-nine  of  the  limb,  a  division  of 
the  vernier  would  be  10''  less  in  value  than  a  division  of  the 
limb  ;  and  the  reading  would  be  within  10"  of  the  exact  position 
of  the  zero  of  the  vernier  on  the  limb. 

But  when  the  highest  degree  of  accuracy  is  sought  for,  as  in 
the  large,  fixed  instruments  of  an  observatory,  the  angle  is  read 
off  by  means  of  the  Beading  Microscope,  instead  of  the  vernier. 

34.  The  Reading  Microscope  is  a  compound  microscope, 
firmly  fixed  opposite  to  the  limb,  and  furnished  with  cross  wires 
in  its  focus,  which  are  movable  by  a  fine-threaded  Micrometer 
Screw.  This  is  a  screw  to  the  head  of  which  is  attached  a 
graduated  cylindrical  head,  that  moves  past  a  fixed  index,  to 
measure,  by  means  of  the  turns  and  parts  of  a  turn  of  the  screw, 
the  exact  distance  through  which  it  is 
moved  in  the  direction  of  its  axis.  In  A 

Fig.  17,  AC  is  the  microscope,  and  MU 
a  portion  of  the  limb  seen  edgewise.  At 
D,  on  the  optical  axis,  is  the  conjugate 
focus  of  the  object-glass  C  ;  when  the 
microscope  is  set  at  the  proper  distance 
from  the  limb,  it  is  coincident  with  the 
focus  of  the  eye-glass  A.  An  image  of  a 
portion  of  the  limb  below  C  is  formed  at 
this  point,  and  is  seen  distinctly  through 
the  eye-glass.  ST  is  a  box  containing 
the  sliding  frame  to  which  the  cross- 
wires  are  attached ;  G,  the  milled  head 
of  the  screw  ;  EF,  the  graduated  cylin- 
drical head,  called  the  graduated  head 
of  the  screw ;  and  i  the  fixed  index. 

The  cross- wires,  with  the  connected 
apparatus  for  giving  motion  to  them,  and  measuring  the  distance 
through  which  they  are  moved,  is  called  a  Micrometer. 

Fig.  18  shows,  upon  an  enlarged  scale,  the  whole  of  the  micro- 
meter, as  it  would  appear  if  viewed  from  A  in  Fig.  17.  aa  is 
the  sliding  frame  to  which  the  cross-wires  are  attached;  c  is  the 
end  of  the  screw  working  into  this  frame ;  and  65,  spiral  springs 
between  the  end  of  the  frame  and  the  end  of  the  box,  to  prevent 
dead  motion  of  the  screw,  and  give  more  steadiness  and  regular- 
ity to  the  movements  of  the  frame  under  the  action  of  the  screw. 

The  divisions  of  the  limb  are  shown  as  short,  heavy,  equidis- 


FIG.  17. 


28 


ASTRONOMICAL   INSTRUMENTS. 


tant  lines.  The  cross- wires  are  the  fine  lines  intersecting  under 
an  acute  angle.  A  wire-pointer,  not  shown  in  the  figure,  in  a 
position  such  that  its  prolongation  would  bisect  this  acute  angle, 


i 

: 

£ 

i_ 

pi 

-2- 
-vi 

a           \      1 

J 

TiTm 

/  \ 

FIG.  18. 

is  generally  used.  On  one  side  of  the  field  is  shown  a  notched 
scale  of  teeth,  called  a  comb  -scale  ;  the  distance  from  the  middle 
of  one  notch  to  the  middle  of  the  next  being  the  same  as  that 
between  the  threads  of  the  screw.  The  wire-pointer  is  moved 
over  this  scale  along  with  the  cross-wires. 

This  scale  is  attached  to  the  micrometer-box,  and  does  not 
move  with  the  cross- wires.  The  number  of  teeth  passed  by  the 
intersection  of  the  wires,  therefore,  shows  the  number  of  turns 
made  by  the  screw  ;  and  the  fractional  part  of  a  turn  is  indicated 
by  the  number  of  divisions  of  the  graduated  head  that  move  past 
the  index  (&',  Fig.  17),  from  the  zero.  If  one  revolution  of  the 
screw  answers  to  a  space  of  V  on  the  limb  of  the  instrument, 
the  number  of  teeth  passed  by  the  intersection  of  the  wires  will 
be  the  number  of  minutes  of  arc  through  which  it  is  moved  ;  and 
if  the  head  of  the  screw  is  divided  into  sixty  equal  parts,  the  line 
of  division  opposite  the  fixed  index  will  give  the  number  of 
seconds  to  be  added  to  the  minutes,  to  determine  the  additional 
space  moved  over. 

In  reading  off  the  angle  the  observer  looks  through  the  micro- 
scope at  the  limb.  The  point  of  intersection  of  the  cross-wires  of 
the  microscope,  when  brought  against  the  central  notch  of  the 
scale,  is  a  fixed  point  of  reference,  like  the  zero  of  a  fixed  vernier- 
plate.  When  the  angle  is  to  be  read,  this  point  will  not,  in 
general,  fall  upon  one  of  the  lines  of  division  of  the  limb.  By 
turning  the  micrometer-screw,  the  intersection  of  the  wires  is 
moved  over  the  space  which  separates  it  from  the  line  of  division 
beyond  which  it  falls ;  the  number  of  teeth  passed  on  the  notched 
scale,  will  then  be  the  number  of  minutes,  and  the  number  of  the 
division  of  the  screw-head  opposite  the  index,  will  be  the  num- 
ber of  seconds,  to  be  added  to  the  angle  taken  from  the  limb. 

To  increase  the  accuracy  of  the  reading,  and  determination  of 
an  angle,  several  microscopes  are  used,  set  opposite  the  limb  at 
equally  distant  points.  The  fraction  of  a  division  in  the  reading 


ASTRONOMICAL    OBSERVATIONS.  29 

is  thus  measured  at  different  points  of  the  circle,  and  the  mean 
of  the  different  measures  is  taken.  Four  reading  microscopes, 
sometimes  six,  or  even  a  greater  number,  are  thus  used.  The 
whole  degrees  and  minutes  are  read  at  only  one  of  the  micro- 
scopes. 

35.  Accuracy  of  Instruments.     It  is  obvious  that,   other 
things  being  the  same,  instruments  are  accurate  in  proportion  to 
the  power  of  the  telescope  and  the.  size  of  the  limb.     The  large 
instruments  now  in  use  in  astronomical  observatories,  are  relied 
upon  as  furnishing  angles  to  within  a  fraction  of  V. 

36.  Time  is  an  essential  element  in  astronomical  observations. 
Three  different  kinds  of  time  are  employed  by  astronomers; 
Sidereal,  Apparent  or  True  Solar,  and  Mean  Solir  Time. 

Sidereal  Time  is  time  as  measured  by  the  diurnal  motion  of  the 
stars;  or,  as  it  is  now  considered,  of  the  vernal  equinox.  A 
Sidereal  Day  is  the  interval  between  two  successive  meridian 
transits  of  a  star;  or,  as  now  defined,  the  interval  between  two 
successive  transits  of  the  vernal  equinox.  It  commences  at  the 
instant  when  the  vernal  equinox  is  on  the  superior  meridian,  and 
is  divided  into  24  Sidereal  Hours. 

Apparent,  or  True  Solar  Time,  is  deduced  from  observations 
upon  the  sun.  An  Apparent  Solar  Day  is  the  interval  between 
two  successive  meridian  passages  of  the  sun's  centre,  commenc- 
ing when  the  sun  is  on  the  superior  meridian.  It  appears  from 
observation  that  it  is  a  little  longer  than  a  sidereal  day,  and  that 
its  length  is  variable  during  the  year.  It  is  divided  into  24  Ap- 
parent Solar  Hours. 

Mean  Solar  Time  is  measured  by  the  diurnal  motion  of  an 
imaginary  sun,  called  the  Mean  Sun,  conceived  to  move  uni- 
formly from  west  to  east  in  the  equator,  with  the  real  sun's  mean 
motion  in  the  ecliptic,  and  to  have  at  all  times  a  right  ascension 
equal  to  the  sun's  mean  longitude.  ^Mean  Solar  Day  com- 
mences when  the  mean  sun  is  on  the  superior  meridian,  and  is 
divided  into  24  Mean  Solar  Hours. 

Since  the  mean  sun  moves  uniformly  and  directly  towards  the 
east,  the  length  of  the  mean  solar  day  must  be  invariable. 

The  Astronomical  Day  commences  at  noon,  and  is  divided  into 
24  hours;  but  the  Calendar  Day  begins  at  midnight,  and  is 
divided  into  two  portions  of  12  hours  each. 

37.  Astronomical  Observations  are,  for  the  most  part, 
made  in  the  plane  of  the  meridian.  But  some  of  minor  import- 
ance are  made  out  of  this  plane.  The  chief  instruments  em- 
ployed for  meridian  observations,  are  the  Meridian  Circk,  and 
the  Transit  Instrument,  used  in  connection  with  the  Astronomical 
Clock.  These  are  the  capital  instruments  of  an  observatory,  in- 
asmuch as  they  serve,  as  will  soon  be  explained,  for  the  deter- 
mination of  the  places  of  the  heavenly  bodies,  which  are  the 
fundamental  data  of  astronomical  science.  The  principal  iustru- 


30  ASTRONOMICAL   INSTRUMENTS. 

ments  used  for  making  observations  out  of  the  meridian  plane, 
are  the  Altitude  and  Azimuth  Instrument,  the  Equatorial,  and  the 
Sextant 


THE  TRANSIT  INSTRUMENT. 

38.  The  Transit  Instrument,  or  Transit,  is  an  instrument  em- 
ployed, in  connection  with  a  clock,  for  observing  the  passage  of 
celestial  objects  across  the  meridian  ;  either  for  the  purpose  of 
determining  their  right  ascension,  or  obtaining  the  correct  time. 
It  is  constructed  of  various  dimensions,  from  a  focal  length  of 
20  inches,  to  one  of  10  feet.  The  larger  and  more  perfect  in- 
struments are  permanently  fixed  in  the  meridian  plane,  and  rest 
upon  stone  piers.  The  smaller  ones  are  mounted  on  portable 
stands.  Fig.  19  represents  a  fixed  transit  instrument  in  its  most 
approved  form.  It  is  a  sketch  of  the  meridian  transit  instru- 
ment of  the  Washington  Observatory,  made  by  Ertel  &  Sons, 
Munich.  The  telescope  has  a  focal  length  of  85  inches,  with 
a  clear  aperture  of  5.3  inches.  TT  is  the  telescope,  firmly  fixed 
to  an  inflexible  axis,  AA,  at  right  angles  to  its  length.  The  axis 
consists  of  two  hollow  cones,  AA,  proceeding  from  the  opposite 
sides  of  a  hollow  cube,  M ;  the  whole  being  cast  in  one  piece. 
The  tube  of  the  telescope  is  composed  of  two  tubes,  which  are 
fastened  by  screws  to  the  other  two  faces  of  the  cube,  M.  The 
axis  terminates  in  two  steel  pivots,  V,  accurately  turned  to  the 
cylindrical  shape,  and  of  equal  size.  These  pivots  rest  on  two 
angular  bearings,  in  form  like  the  upper  part  of  a  Y,  and  called 
Y's.  The  Y's  are  notches  cut  in  two  blocks  of  metal,  set  in 
metallic  boxes ;  the  latter  being  imbedded  in  the  tops  of  the  stone 
piers  PP.  Sufficient  play  is  given  to  the  blocks  in  their  boxes 
to  allow  one  of  the  pivots  to  bs  raised  or  lowered,  and  the  other 
to  be  moved  to  the  right  or  left  by  means  of  adjusting  screws, 
that  give  a  motion  to  the  blocks.  To  relieve  the  pivots  of  a 
portion  of  the  weight  of  the  telescope,  a  brass  pillar,  S,  is  firmly 
set  upon  the  top  of  each  pier,  and  furnishes  a  fulcrum  to  a  lever, 
E,  from  one  end  of  which  depends  a  strong  brass  hook  that 
supports  the  friction  rollers  X,  under  the  end  of  the  axis.  A 
counterpoise,  W,  is  adapted  to  the  other  end  of  the  lever,  which 
serves  to  sustain  the  greater  part  of  the  weight  of  the  telescope, 
and  leaves  only  a  sufficient  pressure  at  the  pivots  to  secure  a 
perfect  contact  with  the  Y's.  This  not  only  saves  the  pivots 
from  wear,  but  gives  the  greatest  possible  freedom  of  motion  to 
the  telescope — the  lightest  touch  of  the  finger  being  sufficient  to 
rotate  the  instrument  upon  the  friction  rollers  on  which  the  axis 
chiefly  rests. 

Illumination  of  ike  Reticle-  Wires.  The  pivots  are  perforated, 
to  admit  the  light  of  a  lamp  placed  on  the  top  of  either  pier. 


THE  TRANSIT  INSTRUMENT. 


31 


FIG.  19. 


32  ASTRONOMICAL   INSTRUMENTS. 

The  light  is  received  upon  a,  plane  metallic  speculum,  sot  within 
the  hollow  cube,  M,  at  a.n  angle  of  45°  to  the  axis  of  the  tele- 
scope, and  is  reflected  to  the  eye-glass;  thus  illuminating  the 
field  of  view,  and  exhibiting  the  wires  of  the  reticle,  at  m,  as  dark 
lines  on  a  comparatively  bright  ground.  The  reflector  has  an 
elliptical  opening  at  its  centre,  to  permit  the  light  that  enters  the 
telescope  from  a  star,  to  pass  on  to  the  eye-glass.  In  observing 
small  stars  the  wires  are  illuminated  from  the  side  of  the  eye- 
glass, by  two  small  lamps  (omitted  in  the  drawing)  suspended 
upon  the  telescope,  near  the  eye-piece,  which  throw  their  light 
obliquely  upon  the  wires,  through  openings  in  the  eye-tube, 
without  illuminating  the  field.  The  wires  are  thus  made  to 
appear  as  bright  lines  on  a  dark  ground.  The  reticle  has  seven 
transit  wires,  placed  at  equal  intervals,  and  two  horizontal 
ones,  between  which  the  star  is  made  to  pass  (30). 

finding  Circles.  On  each  side  of  the  eye-end  of  the  telescope, 
is  fastened  a  small  vertical  graduated  circle,  F,  about  the  centre 
of  which  turns  freely  an  index-arm  which  carries  a  spirit-level 
and  a  vernier.  This  piece  of  apparatus  is  called  a  Finding 
Circle,  or  a  Finder.  An  outline  sketch  of  the  finding  circle,  in 
one  of  its  forms,  is  shown  in  Fig.  20 ;  a  is  the  index-arm,  and  I 


FIG.  20. 

the  level  fastened  at  right  angles  to  this,  at  the  centre  of  the 
divided  circle.  Both  turn  freely  about  this  centre.  At  the 
lower  end  of  a  is  a  vernier,  and  also  a  clamp  and  tangent-screw 
(not  shown  in  the  figure).  The  finding  circles  attached  to  the 
present  instrument  have  a  vernier  at  each  end  of  a  horizontal 
arm  that  carries  the  level ;  and  the  vertical  arm  serves  only  for 
clamping,  and  the  tangent-screw  motion. 

By  means  of  the  finder,  the  telescope  can  be  set  to  any  given 
altitude  or  zenith  distance,  preparatory  to  an  observation  of  the 
meridian  passage  of  a  star.  This  is  done  by  setting  the  vernier 
of  the  finder  to  the  given  angle,  and  then  depressing  the  eye-end 
of  the  telescope  until  the  spirit-level  is  horizontal.  In  accom- 
plishing this,  the  handles,  BB  and  D,  are  used.  The  handle  D 
acts  upon  a  clamp  that  fastens  the  rotation  axis.  When  the 
telescope  has  been  depressed  nearly  to  the  required  position,  it  ia 


ADJUSTMENTS.  33 

clamped  by  this  handle,  and  the  handles  BB,  which  are  connected 
with  tangent-screws,  serve  to  give  the  telescope  a  slow  motion  in 
altitude.  By  the  same  means,  when  the  star  to  be  observed 
enters  the  field  of  view  of  the  telescope,  it  can  be  made  to  pass 
through  the  middle  of  the  field. 

A  Reversing  Apparatus,  or  Car,  with  which  the  instrument 
may  be  lifted  from  the  Y's,  and  the  rotation  axis  reversed,  is 
shown  at  K2.  It  is  mounted  on  grooved  wheels  that  run  upon 
two  rails  laid  in  the  observatory  floor,  between  the  piers  PP. 
The  telescope  having  been  placed  in  a  horizontal  position,  the 
car  is  brought  directly  beneath  the  axis.  By  turning  the  crank 
h,  acting  upon  two  bevelled  wheels,  e  and  f,  the  latter  of  which 
has  an  internal  screw  engaging  in  an  external  screw  upon  the 
lower  end  of  the  vertical  shaft  t,  two  forked  arms,  aa,  are  lifted 
and  brought  into  contact  with  the  axis  at  A  A ;  then,  continuing 
the  motion,  the  telescope  is  lifted  sufficiently  for  the  axis  to  clear 
the  Y's  and  the  friction  rollers  at  XX.  The  car  is  then  rolled 
out  from  between  the  piers,  bearing  the  telescope  with  it ;  the 
instrument  is  turned  half  around  upon  the  vertical  shaft,  the  car 
rolled  back  to  its  former  position,  and  the  axis  lowered  into  the 
Y's.  The  exact  semi-revolution  is  determined  by  the  stop,  d. 

An  observing  couch,  C,  runs  on  the  rails  between  the  piers.  It 
is  so  arranged  that  the  observer,  reclining  upon  it,  may  give  his 
head  any  required  elevation  ;  and  thus  promotes  facility  and 
accuracy  of  observation,  by  giving  greater  steadiness  to  the  head, 
and  relieving  the  observer  of  the  fatigue  of  a  constrained  posi- 
tion when  the  telescope  is  directed  upon  stars  at  high  altitudes. 

L  is  a  striding  level,  which  is  used  in  levelling  the  rotation  axis. 

39.  Adjustments  of  the  Transit.  To  secure  accurate  obser- 
vations with  the  transit,  three  adjustments  of  the  instrument  are 
necessary : 

1.  The  axis  of  rotation  is  to  be  brought  into  a  horizontal  po- 
sition. 

2.  The  line  of  collimation  is  to  be  made  perpendicular  to  the 
axis  of  rotation. 

3.  The  line  of  collimation  is  to  be  brought  accurately  into  the 
meridian  plane. 

When  these  adjustments  have  been  effected,  the  line  of  sight 
will  lie  in  the  plane  of  the  meridian  in  every  position  given  to 
the  telescope. 

40.  First  Adjustment.    The  first  adjustment  is  effected  by 
means  of  the  striding  level,  L,  which  is  applied  to  the  pivots,  YV  ; 
the  feet  of  the  level  having  the  form  of  an  inverted  Y  for  this  pur- 
pose.    By  alternately  working  the  screws  that  raise  or  depress 
one  of  the  pivots,  and  the  adjusting  screws  of  the  spirit-level, 
until  the  level  is  horizontal,  whichever  leg  rests  upon  the  eastern 
end  of  the  axis,  the  axis  may  be  made  truly  horizontal.    Instead 
of  attempting  to  secure  in  this  way  a  perfect  adjustment  of  the 

3 


34  ASTRONOMICAL  INSTRUMENTS. 

axis,  it  is  found  more  convenient  to  determine  the  inclination  of 
the  axis  to  the  horizon,  by  means  of  the  scale  marked  off  upon 
the  tube  of  the  spirit  level,  and  calculate  the  error  that  is  en- 
tailed from  this  cause,  upon  the  observation. 

41.  Second  Adjustment.  The  second,  or  collimation  adjust- 
ment, is  now  generally  made  by  means  of  special  contrivances  for 
the  purpose,  but  it  may  also  be  accomplished  in  the  following 
manner.  Bring  the  telescope  into  a  horizontal  position,  and  direct 
it  upon  a  well-defined  point  of  a  distant  terrestrial  object.  Then, 
by  means  of  the  reversing  apparatus  raise  the  telescope  from  the 
Y's,  and  replace  it  with  the  ends  of  the  axis  reversed.  Bring 
the  telescope  again  into  a  horizontal  position,  and  note  whether 
it  is  directed  upon  the  same  point  as  before.  If  not,  bring  it 
half-way  back  to  this  point  by  the  adjusting  screws  of  the 
reticle,  and  the  remaining  distance  by  the  screws  that  give  a  lateral 
motion  to  one  end  of  the  rotation  axis.  By  one  or  more  repeti- 
tions of  this  process,  the  desired  adjustment  may  be  effected. 

The  better  plan,  and  the  one  ordinarily  adopted  by  astronomi- 
cal observers,  is,  after  the  error  of  collimation  has  been  reduced 
to  a  small  amount,  to  determine  its  value,  and  allow  for  it.  This 
can  readily  be  done  when  the  reticle  is  provided  with  a  mov- 
able micrometer- wire  (31).  It  is  only  necessary  to  measure, 
with  the  micrometer,  the  distance  of  the  point  observed  from  the 
middle  wire  of  the  reticle  in  both  positions  of  the  telescope,  con- 
vert each  of  the  measured  distances,  expressed  in  revolutions  of 
the  screw-head,  into  their  equivalent  angular  measures,  and  take 
the  half  difference  of  the  two  results.  This  will  be  the  error  of 
collimation. 

The  opportunity  of  reversing  the  instrument  also  enables  the 
observer  to  determine  the  correction  for  inequality  of  the  pivots  • 
that  is,  the  inclination  of  the  mathematical  axis  of  rotation  to  the 
horizon  that  may  result  from  any  such  inequality.  This  correc- 
tion is  equal  to  one  quarter  of  the  difference  between  the  inclina- 
tions of  the  line  on  which  the  feet  of  the  level  virtually  rest,  as 
determined  by  the  level,  in  both  positions  of  the  telescope. 

Collimating  Eye-Piece.  The  most  convenient 
method  of  determining  the  error  of  collimation 
is  by  making  a  certain  observation  with  what  is 
called  the  collimating  eye-piece,  substituted  for 
the  ordinary  eye-piece  of  the  telescope  (Fig. 
21).  This  differs  from  the  common  eye-piece 
in  having  an  opening  in  one  side  of  the  tube, 
and  a  metallic  reflector,  of  the  form  of  an  ellip- 
tical ring,  set  obliquely  within  the  tube,  to  re- 
flect the  light  of  a  lamp  upon  the  wires  of  the 
PIG.  21.  micrometer.  The  observation  to  be  made  with 

it  consists  simply  in  looking  vertically  down- 
ward through  the  telescope  at  the  image   of  the  micrometer- 


THIRD  ADJUSTMENT.  35 

wires,  reflected  from  a  basin  of  mercury  placed  on  an  immo- 
vable stone  slab  under  the  telescope.  If  the  axis  has  been 
truly  levelled,  the  error  of  collimation  will  be  half  the  dis- 
tance between  the  middle  wire,  as  seen  directly,  and  its  reflected 
image.  This  distance  can  be  measured  by  means  of  the  mova- 
ble wire  of  the  micrometer.  By  working  the  adjusting  screws 
of  the  reticle,  and  the  vertical  adjusting  screws  of  the  axis  of 
rotation,  the  interval  between  the  wire  and  its  image  may  be 
made  to  disappear  entirely  ;  when  the  axis  will  be  truly  level, 
and  the  line  of  collimation  in  perfect  adjustment. 

42.  Third  Adjustment.  The  piers  must  first  be  established 
in  such  positions  that  the  telescope,  when  the  pivot  ends  of  the 
axis  have  been  placed  in  the  Y's,  and  the  axis  levelled,  will  lie 
nearly  in  the  meridian  plane.  This  may  be  accomplished  by 
bringing  the  telescope,  after  repeated  trials,  into  such  a  position 
that  it  will  be  directed  upon  the  pole-star  when  it  is  on  the  me- 
ridian. By  referring  to  a  map  of  the  stars,  it  may  be  seen  that 
the  pole-star  will  be  nearly  on  the  meridian  when  a  straight 
line  from  it  to  a  point  midway  between  the  fifth  and  sixth 
stars,  designated  as  6  and  £,  in  the  constellation  of  the  Great 
Bear,  is  in  a  vertical  position.  The  pole-star  is  also  known  to 
be  on  the  meridian  when  it  attains  to  its  greatest,  or  least  alti- 
tude. When  the  instrument  has  thus  been  approximately 
established,  it  may  be  more  accurately  adjusted  to  the  meridian, 
with  the  aid  of  the  screws  that  give  a  horizontal  motion  to  one  end 
of  the  axis.  For  this  purpose  observations  may  be  made  upon  the 
pole-star  at  its  upper  and  lower  meridian  transits,  and  the  telescope 
moved  in  azimuth,  until  the  interval  between  the  upper  and  lower 
transit  is  made  equal  to  that  between  the  lower  and  upper  transit. 

The  more  convenient  method  is  to  ascertain  from  existing 
tables  the  time  of  the  meridian  passage  of  some  known  star,  and 
bring  the  middle  wire  of  the  telescope  upon  the  star  at  the 
instant  of  the  transit.  In  order  to  effect  this,  the  error  of  the 
timepiece  must  be  known.  If  it  indicates  sidereal  time,  its  error 
may  be  approximately  determined  with  the  instrument  that  is 
being  established,  by  selecting  a  star  that  passes  the  meridian 
near  the  zenith,  and  noting  the  time  of  its  transit  across  the  mid- 
dle wire  of  the  telescope.  This  time  should  differ  very  little 
from  the  instant  of  the  true  meridian  passage,  as  determined 
from  astronomical  tables ;  the  difference  will  then  be  the  error 
of  the  timepiece,  nearly.  The  subsequent  observations  for 
adjustment  to  the  meridian  plane  should  be  made  upon  stars 
remote  from  the  zenith  (the  pole-star  in  preference).  This  pro- 
cess may  be  many  times  repeated,  until  the  line  of  collimation  of 
the  transit  telescope  is  brought,  with  all  attainable  accuracy,  into 
the  meridian  plane.  Or,  the  error  of  the  adjustment  may  be  cal- 
culated from  the  results  of  the  observations  upon  the  star  near 
the  zenith  and  the  pole-star,  and  allowed  for  in  subsequent  obser- 


^O  ASTRONOMICAL  INSTRUMENTS. 

vations.  This  method  of  adjustment  is  called  the  method  of  high 
and  low  stars.  The  final  result  obtained  by  it  may  be  tested  by 
the  method  of  circumpolar  stars  already  alluded  to  ;  which  has 
the  advantage  of  being  independent  of  the  error  of  the  clock. 

If  the  timepiece  used  in  setting  up  the  transit  keeps  mean  solar 
time,  its  error  may  be  determined  by  measuring  an  altitude  of 
the  sun  with  the  transit  or  sextant,  as  will  hereafter  be  ex- 
plained. 

43.  The  Time  of  the  Meridian   Passage   of  a  Star  is 
ascertained  as  follows :  the  telescope  is  first  set  by  means  of  the 
finding  circle,  to  the  meridian  altitude,  or  zenith  distance  of  the 
star  to  be  observed,  and  the  instants  of  its  crossing  each  of  the 
parallel  wires  of  the  reticle  noted.     The  sum  of  these  observed 
times,  divided  by  the  number  of  the  wires,  will  be  the  time  of 
the  star's  crossing  the  middle  wire  ;  provided  the  wires  are  equi- 
distant.    The  distances  between  the  wires,  in  time,  are  called  the 
wire-intervals.    They  can  be  determined,  and  their  equality  tested, 
by  noting  the  intervals  of  time  employed  by  a  star  situated  on  the 
celestial  equator,  in  passing  over  them  successively;  these  equato- 
rial intervals,  divided  by  the  cosine  of  the  declination  of  any  star, 
will  be  the  wire-intervals  for  that  star.     By  means  of  these  inter- 
vals the  time  of  the  star's  passing  either  wire  can  be  reduced  to 
the  middle  wire.     The  mean  of  such  reduced  times  obtained  for 
all  the  wires,  will  be  the  time  of  the  meridian  transit  of  the  star. 
The  utility  of  having  several  wires,  instead  of  one  only,  will  be 
readily  understood,  from  the  consideration  that  a  mean  result  of 
several  observations  is  deserving  of  more  confidence  than  a  single 
one  ;  since  the  chances  are  that  an  error  which  may  have  been 
made  at  one  observation  will  be  compensated  by  an  opposite 
error  at  another. 

If  the  body  observed  has  a  disc  of  perceptible  magnitude,  as 
in  the  cases  of  the  sun,  moon,  and  planets,  the  time  of  the  pas- 
sage of  both  the  western  and  eastern  limb  across  each  of  the 
parallel  wires  is  noted,  and  reduced  to  the  middle  wire;  the 
mean  of  all  the  results  is  then  taken,  which  will  be  the  instant 
of  the  meridian  transit  of  the  centre.  We  may,  at  the  present 
day,  obtain  the  time  of  the  meridian  passage  of  the  centre  of  the 
sun,  moon,  or  any  planet,  from  an  observation  upon  the  western 
limb  only,  by  adding  "  the  sidereal  time  of  the  semi-diameter 
passing  the  meridian,"  taken  from  the  Nautical  Almanac,  to  the 
observed  time.  Or,  the  observation  may  be  made  upon  the 
eastern  limb,  and  the  same  quantity  subtracted. 

44.  Electro- Chronograph.     The  accuracy  of  transit  observations 
has  recently  been  greatly  increased,  by  the  introduction  of  the 
electro-chronograph.     This  valuable  contrivance  consists  of  an 
electro-magnetic  recording  apparatus,  put  into  communication 
with  the  pendulum  of  an  astronomical  clock,  in  such  a  manner 
that  the  circuit  is  broken  at  a  certain  point  of  each  oscillation  ; 


THE  RIGHT  ASCENSION  OF  A   STAR.  37 

and,  as  a  consequence,  the  seconds  beat  by  the  pendulum  are 
designated  by  a  series  of  equally  distant  breaks  in  a  continuous 
line,  upon  a  roll  of  paper  to  which  an  equable  motion  is  given  by 
machinery. 

The  observer  holds  in  his  hand  a  break-circuit  key,  by  means  of 
which  he  interrupts  the  circuit  at  the  instant  that  the  star  is 
bisected  by  one  of  the  wires  in  the  field  of  the  telescope,  and 
thus  makes  a  break  in  one  of  the  short  lines  that  answer  to  the 
successive  seconds ;  as  shown  between  41s.  and  45s.,  in  Fig.  22. 

40s.          41s.          42s.          43s.          44s.          45s.          46s.          41s.          48s. 


FIG.  22. 

In  this  way,  the  instant  of  the  transit  across  a  single  wire  can  be 
noted  to  within  a  much  smaller  fraction  of  a  second  than  by  the 
common  method.  Besides,  the  number  of  bisections  in  a  single 
culmination  of  a  star,  by  increasing  the  number  of  wires,  may  be 
augmented  fivefold. 

This  method  of  observation  was  adopted  at  the  Washington 
Observatory,  in  1849,  and  soon  after  at  the  Observatory  of  Har- 
vard College.  It  has  since  been  introduced  at  the  Greenwich 
and  other  principal  observatories. 

45.  To  determine  the  Rignt  Ascension  of  a  Star.  When 
a  star  is  on  the  meridian,  its  declination  circle  (def.  16,  p.  15) 
coincides  with  the  meridian ;  moreover,  the  arc  of  the  equator 
which  lies  between  the  declination  circles  of  two  stars,  measures 
their  difference  of  right  ascension.  Thus,  RR'  (Fig.  8)  is  the 
difference  of  right  ascension  of  the  stars  S  and  S' ;  their  absolute 
right  ascensions  being  YR  and  VR/.  In  the  interval  between  the 
transits  of  the  two  stars,  the  arc  RR',  which  is  equal  to  their 
difference  of  right  ascension,  passes  across  the  meridian  at  the 
rate  of  15°  to  a  sidereal  hour.  If,  therefore,  the  times  of  their 
meridian  transits  be  determined  with  the  transit  instrument 
and  sidereal  clock,  the  difference  between  these  times,  converted 
into  degrees  by  allowing  15°  to  the  hour,  will  be  the  difference 
of  right  ascension  of  the  two  stars.  In  this  way,  the  difference 
between  the  right  ascension  of  any  standard  star,  S,  fixed  upon 
as  a  point  of  reference,  and  other  stars,  may  be  successively  de- 
termined. This  having  been  done,  the  absolute  right  ascensions 
of  these  stars  will  become  known  as  soon  as  the  position  of  the 
vernal  equinox  with  respect  to  the  standard  star  has  been  found. 
For,  it  is  plain  that  RR'  being  known,  if  YR  be  also  determined, 
VR7  may  be  found  by  adding  YR  .and  RR'.  The  manner  of 
determining  the  position  of  the  vernal  equinox,  or  the  value  of 
YR,  will  be  explained  in  the  chapter  on  the  Apparent  Motion 
of  the  Sun.  Right  ascensions  are  commonly  expressed  in  time. 


38  ASTRONOMICAL  INSTRUMENTS. 


ASTRONOMICAL  CLOCK. 

46.  The  Astronomical  Clock  is  provided  with  a  pendulum 
so  constructed  that  its  length  is  unaffected  by  changes  of  tempera- 
ture.    The  mercurial   compensation   pendulum,    in   which   the 
ordinary  brass  bob  is  replaced  by  a  glass  jar  containing  a  certain 
quantity  of  mercury,  is  generally  employed.     The  clock  is  secured 
to  a  stone  pier  resting  upon  a  firm  foundation,  which  is  discon- 
nected from  the  floor  of  the  observatory.     It  keeps  sidereal  time. 

47.  To  Regulate  a  Sidereal  Clock.    When  a  clock  is  used 
for  determining  differences  of  right  ascension  (45),  it  is  adjusted  to 
sidereal  time  if  it  goes  equably  and  marks  out  twenty-four  hours 
in  a  sidereal  day  ;    it  being  altogether  immaterial  at  what  time  it 
indicates  Oh.  Om.  Os.     To  ascertain  its  daily  rate,   note   by  the 
clock  the  times  of  two  successive  meridian  transits  of  the  same 
star :   the  difference  between  the  interval  of  the  transits  and 
twenty -four  hours  will  be  the  daily  gain,  or  loss  (as  the  case  may 
be),  of  the  clock  with  respect  to  a  perfectly  accurate  sidereal 
clock.     If  the  gain  or  loss,  when  found  in  this  manner,  proves  to 
be  the  same  each  day,  then  the  mean  rate  of  going  is  the  same 
each  day. 

Error. — The  sidereal  clock  now  in  use  in  astronomical  obser- 
vatories, is  made  to  indicate  Oh.  Om.  Os.  when  the  vernal  equinox 
is  on  the  superior  meridian  ;  and  it  is  necessary  to  know  not  only 
its  rate  but  also  its  error.  This  may  be  found  from  day  to  day 
by  noting  the  time  of  the  transit  of  some  known  star,  whose  place 
has  been  accurately  determined,  and  comparing  this  with  its 
right  ascension  expressed  in  time.  If  the  two  are  equal  the 
clock  is  right ;  otherwise  their  difference  will  be  its  error.  For 
greater  accuracy  in  the  determination  of  the  error  and  rate,  the 
successive  transits  of  several  standard  stars  should  be  noted.  To 
facilitate  these  and  other  determinations,  the  apparent  places  of 
a  large  number  of  stars  are  given  in  nautical  almanacs,  and 
other  similar  works. 

Clock  Stars.  The  stars  most  favorably  situated  for  determin- 
ing the  clock  correction  are  those  which  pass  the  meridian  near 
the  zenith ;  or,  next  to  these,  the  stars  which  cross  the  meridian 
between  the  zenith  and  equator.  Stars  considerably  to  the  north 
of  the  zenith  pass  too  slowly  through  the  field  of  the  telescope; 
and  if  the  transit  instrument  has  not  been  accurately  adjusted 
to  the  meridian,  the  error  in  the  time  of  the  transit  will  be 
greater  in  proportion  as  the  star  observed  is  further  from  the 
zenith. 

4§.  A  Mean  Solar  Clock  is  usually  regulated  by  observations 
upon  the  sun.  The  methods  by  which  its  error  and  rate  are 
determined  will  be  explained  in  the  chapter  on  the  Measurement 
of  Time. 


MERIDIAN  CIRCLE.  39 


MERIDIAN  CIRCLE. 

49.  The  Meridian  Circle  is  an  instrument  used  to  measure  the 
zenith  distance,  or  altitude  of  a  heavenly  body,  at  the  instant  of 
its  arrival  on  the  meridian.     It  is,  in  its  general  construction, 
a  combination  of  the  transit  instrument  and  a  graduated  vertical 
circle  ;  and  is  hence  sometimes  called  the  Transit  Circle.     In  the 
larger  observatories,  it  is  mounted  on  two  piers,  like  the  transit. 
The  graduated  circle  is  firmly  attached  at  right  angles  to  the 
horizontal  axis  of  rotation,  and  turns  with  it.     The  angle  is  read 
from  the  circle  by  a  reading  miscroscope,  attached  to  the  adjacent 
pier ;  or  in  some  instances,  to  a  frame  which  rests  upon  the  axis 
itself.     For  greater  accuracy  four  or  six  reading  microscopes  are 
used,   at  equally   distant  points   of  the   limb.      The   degrees, 
minutes,  and  seconds,  are  read  from  one  of  the  microscopes,  and 
the  seconds  only  from  the  others.     If  the  seconds  read  from 
either  microscope  be  added  to  the  degrees  and  minutes  obtained 
from  the  first,  the  result  will  be  the  reading  of  that  microscope 
reduced  to  the  first.     By  taking  the  mean  of  all  the  results,  for 
the  different  microscopes,  the  errors  from  imperfect  graduation, 
inaccurate  centring,   and  unequal  expansion  of  the  limb,  may 
be  materially  lessened. 

50.  Fig.  23  represents  a   meridian   circle   manufactured   by 
Repsold,  a  celebrated  German  instrument-maker,  and  mounted 
in  1852  in  the  observatory  of  the  United  States  Naval  Academy. 
It  has  two  graduated  circles,  CO  and  C'C',  of  the  same  size,  but 
only  one  of  these,  CO,  is  graduated  finely  ;  this  is  read  by  four 
microscopes,  two  of  which  are  seen  at  RR.     The  microscopes 
are  attached  to  the  four  corners  of  a  square  frame  which  is  centred 
upon  the  rotation  axis ;  but  does  not  turn  with  it,  being  held  in  a 
fixed  position  by  screws  connected  with  the  piers.     Each  hori- 
zontal side  of  the  frame  carries  a  spirit  level,  by  which  any  change 
of  inclination  of  the  frame  with  respect  to  the  horizon  may  be 
detected. 

The  second  circle,  constructed  of  the  same  size  as  the  first, 
far  the  sake  of  symmetry,  is  graduated  more  coarsely,  and  is 
used  only  as  a  finder. 

The  counterpoises  WW  act  at  XX,  to  support  the  greater 
part  of  the  weight  of  the  instrument  upon  friction  rollers,  as  in 
the  case  of  the  transit  instrument.  The  inclination  of  the  rotation 
axis  is  measured  with  a  hanging  level,  LL. 

A  horizontal  arm,  FG,  seen  to  the  right  of  the  telescope  in 
the  figure,  extends  out  from  the  pier,  and  receives  a  vertical  arm 
which  is  connected  with  a  collar  upon  the  rotation  axis.  By 
turning  a  screw,  the  head  of  which  is  at  G,  the  telescope  i& 
clamped  in  the  collar ;  and  then  a  screw  (not  seen  in  the  drawing), 
connected  with  the  arm  FG,  and  acting  horizontally  upon  the 


ASTRONOMICAL  INSTRUMENTS. 


MERIDIAN   CIRCLE.  41 

vertical  arm,  gives  a  fine  motion  to  the  telescope.  FG  turns 
upon  a  joint  at  F;  and  to  the  left  of  the  telescope  is  shown  in 
the  position  it  takes  when  detached  from  the  vertical  arm,  pre- 
paratory to  a  reversal  of  the  instrument. 

Another  arm,  fg,  similar  in  its  form  and  arrangement  to  FG, 
receives  a  vertical  arm  attached  to  the  microscope  frame.  Screws 
connected  with  fg,  and  acting  horizontally  at  g  upon  the  vertical 
arm,  serve  to  adjust  the  frame. 

The  field  is  illuminated  by  light  thrown  into  the  interior  of  the 
telescope  through  tubes  at  AA,  and  reflected  towards  the  reticle 
by  a  mirror  in  the  central  cube.  The  quantity  of  light  is  regu- 
lated by  revolving  discs  with  eccentric  apertures,  at  the  extremi- 
ties of  the  tubes  nearest  the  Y's.  These  discs  are  revolved  by 
means  of  a  cord  to  which  hangs  a  small  weight,  S. 

The  micrometer  at  ra  contains  seven  fixed  transit  threads,  and 
three  equally  distant  horizontal  threads  movable  by  a  micrometer- 
screw.  The  more  common  form  consists  of  a  reticle  with  several 
stationary  transit  wires,  or  threads,  and  one  stationary  horizontal 
wire ;  in  connection  with  one  or  more  movable  horizontal  wires 
(31).  The  movable  micrometer- wires  serve  for  the  measure- 
ment of  small  differences  of  declination. 

51.  Mural  Circle.     This  is  another  form  of  the  meridian 
circle  that  has  been  much  used  in  large  observatories.     The  gra- 
duated limb  of  the  mural  circle  is  secured  to  one  end  of  a  hori- 
zontal axis,  which  is  let  into  a  massive  pier  or  wall  of  stone.  Its 
axis,  therefore,  is  not  symmetrically  supported,  and  it  cannot  be 
reversed.     On  these  accounts  it  is  inferior,  for  nice  determina- 
tions, to  the  form  of  meridian  circle  just  described. 

Mural  circles  have  been  constructed  as  large  as  eight  feet  in 
diameter. 

52.  Adjustments.    The  same  adjustments  have  to  be  effected 
with   the   meridian  circle  as  with  the  transit;    and  the  same 
methods  may  be  adopted.     But  it  is  also  necessary  to  determine 
with  great  accuracy  what  is  called  the  horizontal  point  of  the  limb. 
This  is  the  place  of  the  index,  or  zero  of  the  reading  microscope, 
answering  to  a  horizontal  position  of  the  line  of  collimation  of 
the  telescope. 

53.  To  determine  the  Horizontal  Point  of  tbe  Limb. 
Direct  the  telescope  upon  any  known  star,  at  the  time  of  its  pass- 
ing the  meridian,  and  read  off  the  angle  on  the  limb.   On  the  next 
night,  when  the  star  comes  to  the.  meridian,  direct  the  telescope 
upon  the  image  of  the  same  star  reflected  from  a  basin  of  mer- 
cury, and  note  the  angle  as  before.     By  a  fundamental  law  of 
reflection  the  angle  of  depression  of  this  image  will  be  equal  to 
the  angle  of  elevation  of  the  star.     Accordingly  the  arc  on  the 
limb,  which  passes  before  the  reading  microscope,  in  moving  the 
telescope  from  the  star  to  its  image,  will  be  double  the  altitude  of 
the  star,  and  its  point  of  bisection  the  horizontal  point. 


42  ASTRONOMICAL  INSTRUMENTS. 

This  method  will  not  give  an  exact  result  unless  a  correction  is  applied  for  the 
difference  in  the  values  of  the  atmospheric  refraction  at  the  times  of  observation 
(81).  The  necessity  of  making  this  correction  maybe  avoided,  and  a  more  reliable 
result  obtained,  whenever  the  instrument  is  provided  with  a  micrometer  having  a 
movable  horizontal  wire.  By  a  rapid  manipulation  an  observation  may  then  be 
made  upon  the  star  at  the  time  it  is  crossing  the  first  transit  wire,  and  another 
observation  taken  upon  its  image,  as  it  is  crossing  the  last  transit  wire.  The 
instrument  is  first  set  to  the  altitude  of  the  star,  as  nearly  known,  and  the  cor- 
rection to  this  altitude  measured  by  bringing  the  movable  horizontal  wire  upon 
the  star  at  the  instant  it  is  crossing  the  first  transit  wire.  In  observing  the 
image  of  the  star,  it  is  brought  near  the  fixed  horizontal  wire,  the  limb  clamped, 
and  the  observation  completed  by  the  tangent  screw  of  the  limb.  The  observer 
may  then  read,  at  his  leisure,  the  microscopes  for  the  last  measured  angle,  and  the 
micrometer  correction  to  the  first  angle.  To  each  of  the  angles  measured,  a  small 
correction  must  be  applied  to  reduce  it  to  the  meridian. 

54.  To    measure   tlie   Altitude  of   a    Heavenly  Body. 

(1).  Of  a  fixed  star.  Direct  the  telescope  of  the  meridian  circle 
upon  the  star,  bring  it  on  the  horizontal  wire  of  the  reticle,  and 
clamp  the  limb  ;  then  by  means  of  the  tangent  screw  that  gives  a 
small  motion  to  the  limb  and  telescope,  bisect  the  star  with  the 
horizontal  wire  at  the  instant  of  its  crossing  the  middle  transit  wire. 
Then  read  off  the  angle  from  the  different  microscopes,  as  already 
explained  (49),  and  take  the  mean  of  the  several  results.  This 
must  be  corrected  for  the  deviation  of  the  horizontal  point  from 
the  zero  of  the  limb,  and  all  the  detected  errors  that  result  from 
imperfect  adjustments,  or  defects  of  construction. 

If  an  observation  be  made  upon  the  star  at  the  time  of  its 
crossing  any  other  than  the  central  wire,  it  can  readily  be 
reduced  to  the  meridian. 

(2).  Of  the  sun,  moon,  or  any  planet.  Measure  the  altitudes  of 
the  upper  and  lower  limbs,  and  take  their  half  sum  for  the  alti- 
tude of  the  centre,  or  measure  the  altitude  of  the  upper  or  lower 
limb,  and  add  or  subtract  the  apparent  semi-diameter  of  the 
body,  taken  from  the  Nautical  Almanac.  The  observations  are 
facilitated  by  using  the  movable  micrometer  wire  in  establishing 
the  contact  with  the  limb;  then,  by  turning  the  micrometer 
screw,  measuring  the  interval  between  the  position  of  the 
movable  and  that  of  the  parallel  stationary  wire,  and  adding  this 
measured  interval  to  the  mean  of  the  microscope  readings. 

55.  To  determine  the  Declination  of  a  Heavenly  Body. 
The  meridian  altitude,  or  zenith  distance  of  a  heavenly  body, 
having  been  measured  at  a  place  the  latitude  of  which  is  known, 
its  declination  may  easily  be  found.     For  let  s  (Fig.  10,  p.  21) 
represent  the  point  of  meridian  passage  of  a  star  which  crosses 
to  the  north  of  the  zenith  (Z),  Es  will  be  its  declination  (def.  27, 
p.  17),  Zs  its  meridian  zenith  distance,  and  ZE  the  latitude  of  the 
place  of  observation  (0),  (def.  33,  p.  18)  ;  and  we  obviously  have 

Es= 


If  the  star  cross  the  meridian  at  some  point  s'  between  the 
zenith  (Z)  and  the  equator  (E),  we  shall  have  Es'=ZE—  Zs',  (b)  ; 
and  if  its  point  of  transit  be  some  point  s"  to  the  south  of  the 


ALTITUDE  AND   AZIMUTH  INSTRUMENT.  43 

equator  (E),  we  shall  have  E*"=Z*"— ZE,  and— Es"=ZE— Zs", 
(c).  The  three  formulas  (#),  (5),  and  (c),  may  all  be  compre- 
hended in  one,  viz. : 

Declination = latitude -f- meridian  zenith  distance,  ...  (1) 

If  we  adopt  the  following  conventional  rules :  (1)  north  lati- 
tude is  +,  south  latitude  — ;  (2)  the  zenith  distance  is  north,  or 
south,  according  as  the  star  passes  to  the  north  or  south  of  the 
zenith ;  and  it  has  the  same  sign  as  the  latitude  when  it  has  the 
same  name,  the  contrary  sign  when  it  is  of  a  contrary  name ; 
(3)  north  declination  is  +,  south  declination  — . 

The  latitude  which  is  here  supposed  to  be  known,  may  be 
found  by  measuring  the  meridian  altitudes  of  a  circumpolar  star 
with  the  meridian  circle,  and  taking  their  half  sum.  For,  as  the 
pole  lies  midway  between  the  points  at  which  the  transits  take 
place,  its  altitude  will  be  the  arithmetical  mean,  or  the  half  sum 
of  the  altitudes  of  these  points;  and  the  altitude  of  the  pole  is 
equal  to  the  latitude  of  the  place  (24). 

It  will  be  seen  in  the  next  Chapter,  that  certain  corrections 
must  be  applied  to  all  measured  altitudes. 

56.  To  determine  the  Longitude  and  Latitude  of  a 
Body.    "When  the  right  ascension  and  declination  of  a  heavenly 
body  have  been  obtained  from  observation,  with  a  transit  instru- 
ment and  circle  (45,  55),  its  longitude  and  latitude  may  be  com- 
puted.   For,  let  S  (Fig.  8)  represent  the  place  of  the  body,  VKQE 
the  equator,  VLTW  the  ecliptic,  and  P,  K,  the  north  poles  of  the 
equator  and  ecliptic.     In  the  spherical  triangle  PKS  we  shall 
know  PS  the  complement  of  SR  the  declination,  and  the  angle 
KPS  =  ER=rEY+VR=:900  f  right  ascension;  and  if  we  sup- 
pose the  obliquity  of  the  ecliptic  to  be  known,  we  shall  know 
PK.     We   may  therefore  compute   KS,  and  the  angle  PKS. 
But  KS  is  the  complement  of  SL,  which  is  the  latitude  of  the 
body  S;  and  PKS  =  180°— WKS  =  180°— (WV+  VL)=180° 
—(90°  +  longitude)  =  90°  —longitude. 

The  obliquity  of  the  ecliptic,  which  we  have  here  supposed  to 
be  known,  is,  in  practice,  easily  found ;  for  it  is  equal  to  TQ,  the 
sun's  greatest  declination. 

ALTITUDE  AND  AZIMUTH  INSTRUMENT. 

57.  This  instrument  consists  essentially  of  a  telescope  mounted 
upon  either  a  fixed  or  portable  stand,  and  provided  with  both  a 
vertical  and  a  horizontal  graduated  limb.     The  telescope  turns 
with  the  vertical  limb  about  a  horizontal  axis,  and  the  whole 
turns  about  the  vertical  axis  of  the  horizontal  limb.    The  instru- 
ment is  so  adjusted,  that  when  the  line  of  sight  of  the  telescope 
is  in  the  meridian  plane,  the  zero  of  the  reading  microscope  of 
the  horizontal  limb  will  answer  to  the  zero  of  the  limb,  or  nearly 
so.    If  they  do .  not  correspond,  the  distance  between  them  will 


ASTRONOMICAL   INSTRUMENTS. 


.  24. 


be  the  index  error.  This  having  been  determined,  if  the  tele- 
scope be  directed  upon  a  star  out  of  the  meridian,  the  reading  of 
the  horizontal  limb,  corrected  for  the  index  error,  will  be  the 
azimuth  of  the  star  at  the  instant  of  the  observation.  The  ver- 
tical circle  serves  to  measure  the  altitude.  The  altitude  and 
azimuth  instrument  is  sometimes  called  the  Altazimuth;  also  the 
Astronomical  Theodolite. 

5§.  The  Meridian  L,iiie  (def.  8,  p.  14)  at  a  place  may 
easily  be  determined  with  the  altitude  and  azimuth  instrument, 

by  a  method  called  the  Method 
of  Equal  Altitudes.  Let  0  (Fig. 
24)  represent  the  place  of  ob- 
servation, ISTPZ  the  meridian, 
and  S,  S7  two  positions  of  the 
same  star,  at  which  the  altitude 
is  the  same.  Now,  the  spheri- 
cal triangles  ZPS  and  ZPS' 
have  the  side  ZP  common,  ZS= 
ZS',  and  (allowing  the  stars  to 
move  in  circles)  PS=PS/.  Hence 
they  are  equal,  and  consequently 
the  angle  PZS^PZS';  that  is, 

equal  altitudes  of  a  star  correspond  to  equal  azimuths.  Therefore, 
by  bisecting  the  arc  of  the  horizontal  lirnb,  comprehended 
between  two  positions  of  the  vertical  limb  for  which  the  observed 
altitude  of  a  star  is  the  same,  we  shall  obtain  the  meridian  line. 

The  meridian  line  may  be  approximately  determined,  by  this 
method,  with  the  common  theodolite  ;  the  observations  being 
made  upon  the  sun.  The  result  will  be  more  accurate  if  they 
be  made  towards  the  summer  or  winter  solstice,  when  the  sun 
will  have  but  a  slight  motion  towards  the  north  or  south  in  the 
interval  of  the  observations.  It  is,  however,  easy  to  determine 
and  allow  for  the  effect  of  the  sun's  change  of  place  in  the  heavens. 
When  the  time  is  accurately  known,  the  north  and  south  line 
may  be  found  very  easily  by  directing  the  telescope  of  any 
instrument  that  has  a  motion  in  azimuth,  upon  a  star  in  the  vici- 
nity of  the  pole,  at  the  instant  of  its  arrival  on  the  meridian. 

59.  Zenith  Telescope.  This  may  be  regarded  as  a  modified 
form  of  the  portable  altitude  and  azimuth  instrument.  It  is  of 
great  value  for  the  convenient  and  accurate  determination  of  the 
latitude  of  a  place;  and  has  been  used  for  this  purpose  with 
great  success  in  the  United  States  Coast  Survey.  Its  chief 
peculiarities  consist  irj  the  substitution  of  a  finding  circle  with  a 
delicate  spirit  level,  similar  to  the  finding  circle  of  the  transit 
instrument  (38),  for  the  ordinary  vertical  lirnb  of  the  altitude  and 
azimuth  instrument,  and  the  adaptation  to  the  telescope  of  a  mi- 
crometer with  a  movable  horizontal  wire. 

If  such  a  micrometer  be  adapted  to  a  transit  instrument,  that 


EQUATORIAL.  45 

instrument  may  be  successfully  used  as  a  substitute  for  the 
zenith  telescope,  for  the  accurate  determination  of  the  latitude 
of  a  station.* 

EQUATORIAL. 

60.  The  equatorial  consists  of  a  telescope  mounted  with  two 
axes  of  motion,  at  right  angles  to  each  other,  one  of  which  is 
parallel  to  the  axis  of  the  earth,  and  of  the  celestial  sphere. 
The  angular  movement  about  this  axis  is  measured  by  a  gra- 
duated circular  limb  at  right  angles  to  the  axis,  and  therefore 
parallel  to  the  plane  of  the  equator ;  from  which  the  instrument 
takes  its  name.     This  limb  is  called  the  hour  circle.     There  is 
also  a  graduated  circle,  called  the  declination  circle,  adapted  to 
the  other  axis;  which  lies,  in  every  one  of  its  positions,  in  the 
plane  of  a  celestial  meridian.     The  telescope  turns  in  the  plane 
of  a  celestial  meridian  about  this  axis;  and  can  at  the  same  time 
be  made  to  rotate,  in  connection  with  it,  about  the  other,  or  polar 
axis.    It  can  thus  be  readily  set  upon  any  star,  whose  hour  angle 
and  declination  are  known ;  and  when  once  directed  towards  it, 
can  be  made  to  follow  the  star  in  its  diurnal  motion,  by  simply 
producing  a  continuous  movement  about  the  polar  axis.     This 
motion  is  generally  communicated  by  clock-work,  without  the 
use  of  the  hand. 

Plate  I.  represents  the  large  equatorial  telescope  mounted 
under  the  dome  of  the  observatory  of  Harvard  College.  It  is 
connected  with  a  bed-plate  which  is  fastened  by  screw-bolts  to 
the  top  of  a  granite  block,  in  a  position  parallel  to  the  axis  of  the 
heavens.  This  block  is  ten  feet  in  height,  aud  rests  upon  a 
granite  pier  forty-two  feet  high.  The  clock-work  is  on  the 
further  side  of  the  stone  support,  and  does  not  appear  in  the 
figure.  The  instrument  is  so  nicely  counterpoised  that^it  can  be 
moved  with  the  greatest  ease  by  the  pressure  of  the  hand  upon 
the  end  of  one  of  the  balance  rods. 

61.  U§cs  of  tlie  Equatorial.  A  telescope   thus   equatorially 
mounted,  and  provided  with  a  movable  micrometer- wire,  is  espe- 
cially adapted  to  the  measurement  of  the  apparent  diameter  of  a 
heavenly  body,  the  angular  distance  between  stars  in  close  prox- 
imity, and  in  general  to  all  observations  that  require  the  telescope 
to  be  directed  upon  a  body  for  a  considerable  interval  of  time. 
Accordingly  the  large  telescope  of  every  prominent  observatory 
is  mounted  in  this  manner. 

*  This  has  been  satisfactorily  shown  by  Professor  C.  S.  Lyman,  of  Yale  College 
(see  American  Journal  of  Science,  Vol.  XXX.,  p.  52). 

The  zenith  telescope  is  essentially  the  invention  of  Capt.  Andrew  Talcott,  of 
the  United  States  Corps  of  Engineers,  who  also  devised  a  method  of  determining 
the  latitude  by  this  instrument  which  surpasses  all  others,  both  in  simplicity  and 
accuracy.  This  is  now  known  as  Talcott's  method  (Chauvenet's  Spherical  and 
Practical  Astronomy). 


46 


ASTRONOMICAL  INSTRUMENTS. 


The  equatorial  can  also  be  advantageously  used  for  determin- 
ing the  unknown  place  of  a  fixed  star,  or  planet,  in  the  heavens, 
by  measuring  the  angular  distance  and  direction  of  the  star 
from  some  known  star  seen  with  it  in  the  field  of  the  telescope ; 
or  by  noting  the  interval  of  the  transits,  and  measuring  directly 
the  difference  of  declination  of  the  two  stars.  For  this  purpose 
the  telescope  is  furnished  with  a  certain  form  of  micrometer, 
called  the  Position  Filar  Micrometer  •  with  which  the  measure- 
ments in  question  can  be  made  with  great  accuracy. 

Differences  of  right  ascension  and  declination  can  also  be  mea- 
sured with  the  equatorial,  by  means  of  the  hour  and  declination 
circles,  but  with  much  less  accuracy  than  with  the  transit  instru- 
ment and  meridian  circle. 

62.  Position  Filar  Micrometer.  This  piece  of  apparatus  serves  at  the  same  time 
to  measure  small  angular  distances,  and  the  angle  included  between  the  line  connect- 
ing two  stars  in  close  proximity  and  the  celestial  meridian.  This  angle  is  called 
the  angle  of  position  of  one  of  the  stars  with  respect  to  the  other.  It  is  estimated 
from  the  S.  round  by  the  "W.  to  360°.  The  Filar  Micrometer,  designed  for  the 
measurement  of  small  angles,  is  shown  in  Fig.  25.  It  is  the  same  in  principle  as 
the  micrometer  employed  in  the  reading  microscope  (34). 


FIG.  26. 


FIG.  25. 

It  consists  of  two  forks  of  brass,  bb'b,  cc'c,  sliding  within  a  rectangular  brass 
box,  aa  a,  and  one  within  the  other.  Each  of  these  forks  carries  a  very  fine  wire, 
r  spider  line,  stretched  perpendicularly  across  from  one  prong  to  the  other;  they 
are  movable,  and  the  parallel  wires  which  they  carry,  by  micrometer  screws 
passing  through  the  ends  of  the  box,  and  attached  to  the  forks.  A  third  an* 
stationary  wire,  Z,  perpendicular  to  the  other  two,  is  attached  to  a  diaphragm  dis- 
connected from  the  forks.  The  heads  of  the  screws  are  not  shown  in  the  figure, 

t  they  may  be  seen  in  Fig.  26,  in  which  b  is  the  micrometer-box.     The  eye- 

se  is  screwed  into  the  micrometer-box,  as  shown  in  Fig.  26.     The  graduated 

screw-heads  are  connected  with  nuts  which  turn,  without  advancing,  upon  the 

screws  that  are  fastened  to  the  forks.     Accordingly  by  turning  the  nuts,  the  forks 

may  be  moved  either  forwards  or  backwards. 

A  stationary  comb-scale  on  one  side  of  the  box,  indicates  the  number  of  revolu- 


EQUATORIAL.  47 

tions  of  either  screw,  answering  to  any  distance  that  the  wires  may  be  separated 
from  each  other ;  and  the  fractional  part  of  a  revolution  is  shown  by  the  gradu- 
ated head  of  the  screw.  The  value  of  one  revolution  of  the  micrometer-screw 
may  be  found  by  bringing  the  two  parallel  wires  into  a  position  perpendicular  to 
the  celestial  equator,  separating  them  by  a  certain  number  of  revolutions,  and 
then  noting  the  time  taken  by  an  equatorial  star  to  traverse  the  interval  between 
them.  The  interval  of  time  thus  obtained,  converted  into  the  equivalent  angular 
space  by  allowing  15"  to  1s,  will  be  the  number  of  seconds  of  arc  answering  to 
the  assumed  number  of  revolutions  of  the  screw. 

To  adapt  the  filar  micrometer  to  the  measurement  of  angles  of  position,  the  mi- 
crometer-box, with  its  attached  eye-piece,  is  so  mounted  as  to  admit  of  a  rotation 
around  the  centre  of  a  graduated  circle  (Fig.  26).  The  circle  is  fastened  at  the 
end  of  the  reticle-tube,  and  in  a  plane  perpendicular  to  the  optical  axis  of  the 
telescope.  The  revolving  motion  is  produced  by  a  milled-head  screw  s,  which 
works  on  an  interior  toothed  wheel;  and  the  angle  is  read  off  upon- the  stationary 
graduated  circle,  by  aid  of  the  vernier  movable  with  the  plate  a. 


SEXTANT, 

63.  The  instruments  which  have  now  been  described  are  ob- 
servatory instruments,  the  chief  design  of  whose  construction  is 
to  furnish  the  places  of  the  heavenly  bodies  with  all  attainable 
exactness.  That  of  which  we  are  now  to  treat  is  much  less  exact, 
though  still  of  great  utility  in  effecting  certain  important  astro- 
nomical determinations  ;  as  of  the  latitude  or  longitude  of  a  place, 
and  the  time  of  day.  It  is  chiefly  used  by  navigators,  and 
astronomical  observers  on  land,  who  are  precluded  by  their 
situation,  or  other  circumstances,  from  using  the  more  accurate 
instruments  of  an  observatory.  It  is  much  more  conveniently 
portable  than  any  of  these,  and  has  not  to  be  set  up  and  adjust- 
ed at  every  new  place  of  observation.  Besides,  as  it  is  held  in 
the  hand,  it  can  be  used  at  sea,  where  by  reason  of  the  agitations 
of  the  vessel,  no  instrument  supported  in  the  ordinary  way  is  of 
any  service. 

61.  Construction :— Principle  of  Construction.  The 
sextant  may  be  defined,  in  general  terms,  to  be  an  instrument 
which  serves  for  the  direct  admeasurement  of  the  angular 
distance  between  any  two  visible  points.  The  particular  quan- 
tities that  may  be  measured  with  it,  are ;  1st,  the  altitude  of 
a  heavenly  body ;  2d,  the  angular  distance  between  any  two 
visible  objects  in  the  heavens  or  on  the  earth.  Its  essential 
parts  are  a  graduated  limb  BC  (Fig.  27),  comprising  about  60 
degrees  of  the  entire  circle,  which  is  attached  to  a  triangular 
frame  BAG ;  two  mirrors,  of  which  one  (A)  called  the  Index 
Glass,  is  movable  in  connection  with  an  index,  G,  about  A,  the 
centre  of  the  limb,  and  the  other  (D)  called  the  Horizon  Glass, 
is  permanently  fixed  parallel  to  the  radius  AC  drawn  to  the 
zero  point  of  the  limb,  and  is  only  half  silvered  (the  upper 
half  being  transparent) ;  and  a  small  immovable  telescope  at 
E,  directed  towards  the  horizon-glass.  The  principle  of  the 
construction  and  use  of  the  sextant  may  be  understood  from 


48 


ASTRONOMICAL  INSTRUMENTS. 


what  follows :  A  ray  of  light  SA  from  a  celestial  object  S, 
which  impinges  against  the  index-glass,  is  reflected  off  at  an 
equal  angle,  and  striking  the  horizon-glass  (D)  is  again  reflected 
to  E,  where  the  eye  likewise  receives  through  the  transparent 


FIG.  27. 

part  of  that  glass  a  direct  ray  from  another  point  or  object  S'. 
jSTow,  if  AS'  be  drawn,  directed  to  the  object  S',  SAS',  the 
angular  distance  between  the  two  objects  S  and  S',  is  equal 
to  double  the  angle  GAG-  measured  upon  the  limb  of  the  in- 
strument (AC  being  parallel  to  the  horizon-glass).  For,  when 
the  index-glass  is  parallel  to  the  horizon-glass,  and  the  angle 
on  the  limb  is  zero,  AD,  the  course  of  the  first  reflected  ray, 
will  make  equal  angles  with  the  two  glasses,  and  therefore  the 
angle  SAD  will  become  the  angle  S'AD,  (= ADE ;)  and  the 
observer,  looking  through  the  telescope,  will  see  the  same  ob- 
ject S'  both  by  direct  and  reflected  light.  Now,  if  the  index- 
glass  be  moved  from  this  position  through  any  angle,  CAGr,  the 
angle  made  by  the  reflected  ray  which  follows  the  direction  AD, 
with  this  glass,  will  be  diminished  by  an  amount  equal  to  this 
angle ;  for,  we  have  DAG^DAC— CAGr.  Therefore  the  angle 
made  with  the  index-glass  by  the  new  incident  ray  SA,  which 
after  reflection  now  pursues  the  same  course  ADE,  and  reaches 
the  eye  at  E,  as  it  is  always  equal  to  that  made  by  the  re- 
flected ray,  will  be  diminished  by  this  amount.  Consequently, 
the  incident  ray  in  question  will  on  the  whole,  that  is,  by  the 
diminution  of  its  inclination  to  the  mirror  by  the  angle  CAGr, 
and  by  the  motion  of  the  mirror  through  the  same  angle,  be 
displaced  towards  the  right,  or  upwards,  an  angle  S'AS  equal  to 
2GAC.  Thus,  the  angular  distance  SAS'  of  two  objects  S,  S', 
seen  in  contact,  the  one  (S')  directly,  and  the  other  (S)  by  reflec- 


THE  SEXTANT.  •  •    .  49 

tion  from  the  two  mirrors,  is  equal  to  twice  the  angle  CAGr  that 
the  index-glass  is  moved  from  the  position  (AC)  of  parallelism 
to  the  horizon -glass. 

Hence  the  limb  is  divided  into  120  equal  parts,  which  are 
called  degrees ;  and  to  obtain  the  angular  distance  between  two 
points,  it  is  only  necessary  to  sight  directly  at  one  of  them, 
and  then  move  the  index  until  the  reflected  image  of  the  other 
is  brought  into  contact  with  it ;  the  angle  read  off  on  the  limb 
will  be  the  angle  sought. 

To  obtain  the  angular  distance  between  two  bodies  which 
have  a  sensible  diameter,  bring  the  nearest  limbs  into  contact, 
and  to  the  angle  read  off  on  the  limb  add  the  sum  of  the  appar- 
ent semi-diameters,  of  the  two  bodies,  or  bring  the  farthest  limbs 
into  contact,  and  subtract  this  sum. 

65.  The  Detail  of  the  Construction  of  the  Sextant  is 
shown  in  Fig.  28.  The  limb,  and  the  triangular  frame  to  which  it  is 


attached,  are  of  hammered  brass,  and  strengthened  by  cross-plates. 
The  graduation  is  upon  silver  inlaid  in  the  brass.  Each  degree  is 
divided  into  six  equal  parts,  of  10'.  N  is  the  horizon-glass,  fastened 
to  the  frame  in  the  position  before  stated  ;  I  the  index-glass,  in  a 
brass  frame,  attached  to  the  index-bar  CD,  by  the  screws  sss, 
and  movable  with  it  about  the  centre  C  of  the  graduated  arc. 
These  two  mirrors  are  of  plate-glass  silvered.  The  upper  half 


50  ASTKONOMICAL  INSTRUMENTS. 

of  the  horizon-glass  is  left  unsilvered,  that  the  direct  rays  from 
the  object  towards  which  the  small  telescope,  T,  is  directed  may 
not  be  intercepted.  The  telescope  is  supported  in  a  ring,  K, 
attached  to  a  stem  underneath,  which  can  be  raised  or  lowered 
by  a  screw.  By  this  means  the  relative  brightness  of  the  direct 
and  reflected  images  can  be  regulated.  M  is  a  microscope,  mov- 
able about  a  centre  on  the  index-bar,  used  in  reading  the  angle 
from  the  vernier  at  D.  The  vernier  is  so  divided  as  to  give  the 
angle  to  within  10".  At  B,  under  the  index-bar,  is  a  screw  for 
clamping  it  to  the  limb  ;  and  G  is  a  tangent  screw  for  giving  the 
bar,  with  the  index-glass,  a  small  motion,  in  securing  the  accurate 
contact  or  coincidence  of  the  images.  H  is  a  wooden  handle  at 
the  back  of  the  sextant,  by  which  it  is  held  when  an  observa- 
tion is  taken.  At  E  and  F  are  colored  glasses  of  different 
shades,  to  diminish  the  intensity  of  the  light  when  the  sun  is 
observed.  Those  at  F  are  interposed  between  the  index-glass 
and  the  horizon-glass  when  the  sun  is  seen  by  reflection  from  the 
index-glass.  The  others  are  used  when  the  telescope  is  directed 
upon  the  sun. 

66.  Adjustments.  The  adjustments  of  the  sextant  consist  in  setting  the 
index-glass  and  the  horizon-glass,  and  bringing  the  line  of  sight  of  the  telescope 
parallel  to  the  plane  of  the  graduated  arc,  and  in  determining  the  index  error. 

The  index-glass  may  be  adjusted  by  setting  the  index  near  the  middle  of  the  arc, 
placing  the  eye  nearly  in  the  plane  of  the  sextant,  and  near  the  index-glass,  and 
observing  whether  the  arc  seen  directly  and  its  reflected  image  form  one  continu- 
ous arc.  If  the  reflected  image  does  not  appear  to  form  a  true  continuation  of  the 
arc,  the  index-glass  is  not  perpendicular  to  the  plane  of  the  sextant.  It  may  be 
corrected  by  loosening  the  screws  s  s  s,  and  inserting  a  piece  of  paper  under  the 
plate  through  which  they  pass. 

The  horizon-glass  is  adjusted  by  sighting  through  the  telescope  at  a  star,  and  mov- 
ing the  index  until  the  direct  and  reflected  image  of  the  star  pass  each  other.  If,  in 
passing,  the  two  images  can  be  made  to  coincide,  the  horizon-glass  is  perpendicular 
to  the  plane  of  the  instrument.  If  any  correction  is  necessary,  it  can  be  made  by 
turning  a  small  screw  at  the  top  or  bottom  of  the  horizon-glass. 

To  test  the  position  of  the  line  of  sight  of  the  telescope,  select  two  objects,  as  two 
stars,  100°  to  120°  apart,  and  bring  the  reflected  image  of  the  one  in  contact 
with  the  direct  image  of  the  other,  on  the  wire  within  the  telescope  that  is  near- 
est the  plane  of  the  sextant :  if  then,  on  moving  the  instrument,  the  contact  re- 
mains when  the  images  are  thrown  upon  the  other  parallel  wire  of  the  telescope 
(although  a  separation  occurs  in  the  interval  between  them),  no  adjustment  is  re- 
quired. It  can  be  made,  when  necessary,  by  means  of  two  small  screws  in  the 
ring  which  supports  the  telescope. 

To  find  the  index  error.  Bring  the  direct  and  reflected  images  of  the  same  point 
of  a  distant  terrestrial  object,  or  of  the  same  star,  into  coincidence,  and  read  off  the 
arc.  This  reading  will  be  the  index  error,  and  may  be  either  positive  or  negative. 

6Y.  Taking  an  Angle.  When  observing  with  the  sextant, 
it  is  held  in  the  right  hand  by  the  handle,  and  the  telescope 
directed  upon  one  of  the  two  objects  whose  angular  distance  is 
to  be  measured,  generally  the  fainter  one.  It  is  then  turned 
about  the  line  of  sight  until  the  other  object  lies  in  its  plane ; 
and  the  index  moved  with  the  left  hand  until  the  reflected 
image  of  this  object  is  brought,  at  the  centre  of  the  field  of  the 
telescope,  into  apparent  contact  with  the  object  seen  directly ;— « 


THE    SEXTANT.  51 

the  contact  being  finally  effected  by  the  use  of  the  tangent  screw. 
The  angle  is  then  read  from  the  limb  and  vernier,  with  the  mi- 
croscope. 

When  the  sextant  is  employed  to  take  the  altitude  of  a  heavenly 
body,  a  horizontal  reflector,  called  an  Artificial  Horizon,  is  placed 
in  front  of  the  observer.  The  angle  between  the  body  and  its 
reflected  image  is  then  measured  as  if  this  image  were  a  real 
object;  the  half  of  which  will  be  the  altitude  of  the  body. 

A  small  quantity  of  mercury,  poured  into  a  shallow  vessel 
of  tinned  iron  or  copper,  forms  a  very  good  artificial  horizon. 

In  obtaining  the  altitude  of  a  body  at  sea,  its  altitude  above 
the  visible  horizon  is  measured,  by  bringing  the  lower  limb  into 
contact  with  the  horizon.  To  this  angle  is  added  the  apparent 
semi-diameter  of  the  body,  and  from  the  result  is  subtracted  the 
depression  of  the  visible  horizon  below  the  horizontal  line,  called 
the  Dip  of  the  Horizon. 

68.  Hartley's  Quadrant.     Hadley's  Quadrant  differs  from 
the  sextant  in  having  a  graduated  limb  of  45°,  instead  of  60°,  in 
real  extent,  and  a  sight- vane  instead  of  a  small  telescope.     It  is 
not  capable,  then,  of  measuring  any  angle  greater  than  about  90°, 
while  the  sextant  will  measure  an  angle  as  great  as  120° ;  or 
even  140°  (for  the  graduation  generally  extends  to  140°).     The 
quadrant  is  also  inferior  to  the  sextant  in  respect  to  materials 
and  workmanship,  and  its  measurements  are  less  accurate. 

69.  Reflecting  Circle.     The  ^Reflecting  Circle  is  but  an  en- 
larged sextant.     Its  limb  is  a  full  circle,  and  the  index-arm  is 
prolonged  in  the  other  direction,  and  carries  a  vernier  on  each 
end.     The  angle  is  read  from  each  vernier,  and  the  mean  of  the 
two  readings  taken,  to  eliminate  the  error  of  eccentricity. 

70.  Prismatic  Sextant.     This  is  an  improved  form  of  sex- 
tant, recently  introduced.     It  takes  its  name  from  the  fact  that 
a  reflecting  prism  is  used  in  place  of  the  ordinary  horizon-glass. 
This  prism  also  occupies  a  different  position  with  respect  to  the 
index-glass.     The  graduated  limb  extends  120°.     The  prismatic 
sextant  can  be  used  to  measure  an  angular  distance  of  180°,  and 
an  altitude  of  90°.     It  is  also  superior  to  the  ordinary  sextant 
in  certain  other  peculiarities  of  construction. 

Prismatic  Reflecting  Circles  are  also  constructed  which  possess 
similar  advantages  over  the  ordinary  reflecting  circle. 

ERRORS  OF  INSTRUMENTAL  ADMEASUREMENT. 

Tl.  Whatever  precautions  may  be  taken,  the  results  of  instru- 
mental admeasurement  will  never  be  wholly  free  from  errors. 
Errors  that  arise  from  inaccuracy  in  the  workmanship  or  ad- 
justment of  the  instrument,  may  be  detected  and  allowed  for. 
But  errors  of  observation  are,  obviously,  undiscoverable.  Since, 
however,  the  chances  are,  that  an  error  committed  at  one  obser- 


52  ASTRONOMICAL  INSTRUMENTS. 

vation,  will  be  compensated  by  an  opposite  error  at  another,  it  is 
to  be  expected  that  a  more  accurate  result  will  be  obtained  if  a 
great  number  of  observations,  under  varied  circumstances,  be 
made,  instead  of  one,  and  the  mean  of  the  whole  taken  for  the 
element  sought.  And  accordingly,  it  is  the  uniform  practice  of 
astronomical  observers  to  multiply  observations  as  much  as  is 
practicable. 

72.  Instrumental  Errors  may  be  divided  into  three  classes ;  viz.  errors 
of  construction,  errors  of  adjustment,  and  incidental  errors.     Errors  of  construction, 
in  the  best  instruments,  result  chiefly  from  imperfect  graduation^  an  eccentricity 
of  the  limb,  an  inequality  or  an  elliptidtyof  the  pivots,  and  an  imperfect  rigidity  of  the 
telescope  or  axis.  The  effect  of  eccentricity  and  of  the  ellipticity  of  the  pivot,  may  be 
eliminated  by  taking  the  mean  of  the  readings  of  two  microscopes,  at  opposite 
points  of  the  limb.     The  error  of  graduation  may  be  greatly  reduced,  by  reading 
the  angle  from  several  equidistant  points  of  the  limb,  and  taking  the  mean  of  all 
the  readings.     When  the  construction  of  the  instrument  is  such  that  the  principle 
of  repetition  may  be  adopted — that  is,  the  angle  read  off  from  all  parts  of  the  limb 
— the  error  of  graduation  may,  theoretically  speaking,  be  removed  entirely. 

It  is  not  the  practice  of  astronomical  observers  to  strive  to  bring  instruments 
into  the  nicest  possible  adjustment,  but  instead,  after  a  good  adjustment  has  been 
effected,  to  deduce,  by  a  systematic  series  of  observations,  the  several  errors  that 
remain,  and  derive  from  these  the  corrections  to  be  applied  to  the  quantity  to  be 
determined. 

Incidental  errors  may  arise  from  diverse  effects  produced  by  changes  of  temper- 
ature, especially  an  unequal  expansion  of  different  parts  of  the  limb,  and  a  derange- 
ment of  the  microscopes ;  from  flexure  produced  by  weight ;  and  also  from  vibra- 
tions produced  by  passing  vehicles,  and  other  derangements  from  extraneous 
mechanical  causes.  All  such  errors  may  be  mostly  neutralized  by  making  nume- 
rous measurements,  under  a  great  variety  of  circumstances. 

THE  TELESCOPE. 

73.  An  observatory  is  not  completely  furnished  unless  it  is  supplied  with  a  large 
telescope  for  examining  the  various  classes  of  objects  in  the  heavens ;  and  one  or 
more  smaller  ones  for  exploring  the  heavens  and  searching  for  particular  objects 
invisible  to  the  naked  eye,  as  faint  comets,  and  making  observations  upon  occa- 
sional celestial  phenomena,  as  eclipses  of  the  sun  and  moon,  occupations  of  the  stars, 
etc.    Telescopes  are  divided  into  the  two  classes  of  Reflecting  and  Refracting  Tele- 
scopes.   In  the  former  class,  the  image  of  the  object  is  formed  by  a  concave  specu- 
lum, and  in  the  latter  by  a  converging  achromatic  lens.    This  image  is  viewed  and 
magnified  by  an  eye-glass ;  or  rather  by  an  achromatic  eye-piece  consisting  of  two 
glasses.    In  the  simplest  form  of  the  reflecting  telescope,  the  Herschelian,  the 
image  formed  by  the  concave  speculum  is  thrown  a  little  to  one  side,  and  near  the 
open  mouth  of  the  tube,  where  the  observer  views  it  through  the  eye-glass,  with 
his  back  turned  towards  the  object. 

74.  Magnifying  power — illuminating  power — space-penetrating  power.    The  magni- 
fying power  of  a  telescope  is  to  be  carefully  listinguished  from  its  illuminating,  and 
space-penetrating  power.    A  telescope  magnifies  by  increasing  the  angle  under 
which  the  object  is  viewed ;  it  increases  the  light  received  from  objects,  and  reveals 
to  the  sight  remote  stars,  nebulae,  etc.,  by  intercepting  and  converging  to  a  point 
a  much  larger  beam  of  rays.     The  magnifying  power  is  measured  by  the  ratio  of 
the  focal  length  of  the  object-glass,  or  speculum,  to  that  of  the  eye-piece.     The 
'illuminating  power,  by  which  it  reveals  stars  invisible  to  the  naked  eye,  if  we  leave 
out  of  view  the  amount  of  light  lost  by  reflection  and  absorption,  is  measured  by 
the  proportion  which  the  area  of  the  object-glass,  or  speculum,  bears  to  that  of  the 
pupil  of  the  eye.     Since  the  quantity  of  light  received  from  any  luminous  point, 
viewed  at  different  distances  by  the  naked  eye,  decreases  in  the  same  proportion 
that  the   square  of  the  distance  increases,  and  the  quantity  of  light  from  tho 
same  point,  conveyed  to  the  eye  by  a  telescope,  is  augmented  in  the  ratio  of  the 


THE  TELESCOPE.  53 

square  of  the  diameter  of  its  aperture  to  the  square  of  the  diameter  of  the  pupil  of 
the  eye,  it  follows  that  the  diminution  of  the  light  from  an  increase  of  distance, 
will  be  just  supplied  if  the  aperture  of  the  telescope  exceed  in  its  diameter  that  of 
the  pupil  of  the  eye  in  the  same  ratio  that  the  distance  is  augmented.  The  power 
of  a  telescope  to  penetrate  into  space,  and  discern  stars,  therefore,  exceeds  that  of  the 
naked  eye  in  the  same  ratio  that  the  diameter  of  its  aperture  exceeds  that  of  the 
pupil  of  the  eye  (0.2  in.).  In  the  larger  reflecting  telescopes,  the  space-penetrat 
ing  power,  calculated  by  this  rule,  requires  to  be  diminished  about  one-filth,  in 
consequence  of  the  loss  of  light  incident  to  the  use  of  the  telescope. 

Telescopes  are  provided  with  several  eye-glasses,  of  various  powers.  The  power 
to  be  used  varies  with  the  object  to  be  viewed,  and  the  purity  and  degree  of  tran- 
quillity of  the  atmosphere.  Of  two  telescopes  of  the  same  focal  length,  that 
which  has  the  largest  aperture  will  form  the  brightest  image  in  the  focus,  and 
therefore,  other  things  being  equal,  admit  of  the  use  of  the  most  powerful  eye- 
piece. In  this  way,  it  happens  that  the  available  magnifying  power  indirectly 
depends  materially  upon  the  size  of  the  aperture.  In  all  telescopes,  there  is  a  cer- 
tain fixed  ratio  between  the  aperture  and  focal  length,  or  at  least  limit  to  this 
ratio.  In  reflecting  telescopes,  it  is  one  linear  inch  of  aperture  for  every  foot  of 
focal  length,  and  in  refracting  telescopes  one  inch  of  aperture  for  from  one  to  two 
feet  of  focal  length.  Reflectors  and  refractors  of  the  same  focal  length,  have  about 
the  same  actual  magnifying  and  illuminating  power. 

The  highest  theoretical  magnifying  power  that  has  yet  been  obtained  is  about 
7,000.  But  the  highest  actually  available  power,  in  observing  any  celestial  object, 
does  not  exceed  2,500.  The  higher  powers  can  be  used  only  upon  double  stars, 
and  clusters  of  stars.  With  the  best  telescopes,  a  magnifying  power  of  four  or  five 
hundred  is  the  highest  that  can  be  applied  to  the  moon  and  planets ;  owing  to  the 
great  diminution  of  brightness  that  results  from  the  enlargement  of  the  image. 

7  5.  Defining  power.  Telescopes  of  equal  size  may  differ  materially  in  their  defin- 
ing power:  that  is,  in  their  capability  to  show  the  planets,  and  other  celestial 
objects  which  have  a  sensible  disc,  with  a  sharp  outline,  and  all  their  peculiarities 
of  appearance  with  distinctness,  and  to  separate  close  double  stars  and  clusters  of 
stars.  The  excellence  of  telescopes  in  this  respect,  depends  upon  the  precision  of 
form  and  perfection  of  polish  of  the  lenses,  their  freedom  from  chromatic  and 
spherical  aberrations,  and  other  niceties  of  construction. 

76.  The  field  of  view  of  telescopes  diminishes  in  proportion  as  the  magnifying 
power  increases.     It  is  stated  that  with  a  magnifying  power  of  between  100  and 
200  it  is  a  circle  not  as  large  as  the  full  moon ;  and  with  a  power  of  600  or  1,000 
is  nearly  filled  by  one  of  the  planets,  while  a  star  will  pass  across  it  in  from  two 
to  three  seconds. 

The  diminution  of  the  field  of  view,  and  the  trepidations  of  the  image  occasioned 
by  the  varying  density  of  the  atmosphere,  and  the  unavoidable  tremors  of  the 
instrument,  must  ever  affix  a  practical  limit  to  the  magnifying  power  of  telescopes. 
This  limit,  it  is  probable,  is  already  nearly  attained ;  for  the  highest  powers  of  the 
best  telescopes  can  now  be  used  only  in  the  most  favorable  states  of  the  weather. 
The  illuminating  and  space-penetrating  power  of  telescopes  may,  however,  yet  be 
materially  increased,  and  a  greater  distinctness  and  definiteness  in  the  outline  of 
objects  obtained. 

77.  Large  Telescopes.    The  largest  reflecting  telescope  that  has  yet  been  con- 
structed and  directed  to  the  heavens,  is  the  great  Rosse  Telescope,  devised  and  con- 
structed by  Lord  Rosse,  of  Ireland.     It  has  a  focal  length  of  53  feet,  and  an  aper- 
ture of  6  feet.     Its  illuminating  power  is  about  78,000;  and  its  space-penetrating 
power,  for  single  stars,  about  280  times  the  distance  of  the  most  remote  star  visible 
to  the  naked  eye.     The  most  powerful  refractor  yet  constructed,  is  the  great  Clark 
Tdescope,  made  by  Clark  &  Sons,  Cambridgeport,  Mass.,  and  recently  set  up  in  the 
Chicago  Observatory.     It  has  a  clear  aperture  of  18£  inches,  and  a  focal  length  of 
23  feet.     It  has,  by  the  adaptation  of  different  eye-pieces,  different  magnifying 
powers,  varying  from  70  to  about  2,000.     The  great  telescope  of  the  Observa- 
tory of  Harvard  College  has  an  aperture  of  15  inches,  and  a  focal  length  of  22£  feet. 
Its  highest  magnifying  power  is  2,000.     The  refractor  of  the  observatory  at  Pul- 
kova,  hi  Russia,  is  but  slightly  inferior  to  this  in  its  dimensions  and  capabilities. 
Refracting  telescopes  of  large  dimensions  and  great  excellence,  are  mounted  equa- 
torially  in  all  the  prominent  observatories  m  the  United  States  and  in  Europe. 


54  COKRECTIONS  OF   MEASURED  ANGLES. 


CHAPTER  IV. 

CORRECTIONS  OF  MEASURED  ANGLES. 

78.  Angles  measured  at  the  earth's  surface  with  astronomical 
instruments  answer  to  the  Apparent  Place  of  a  heavenly  body, 
and  are  termed  Apparent  elements.     In  astronomical  language 
the  True  Place  of  a  heavenly  body  is  its  real  place  in  the  heav- 
ens, as  it  would  be  seen  from  the  centre  of  the  earth.     Angles 
which  relate  to  the  true  place  are  denominated  True  elements. 
The  co-ordinates  of  the  apparent  place  of  a  body  are  termed  its 
apparent  co-ordinates,  and  those  of  its  true  place  its  true  co-ordi- 
nates. 

79.  Corrections.     The  apparent  co-ordinates  are  reduced  to 
the  true,  by  the  application  of  certain  corrections,  called  Refrac- 
tion, Parallax,  and  Aberration.    Refraction  and  aberration  are  cor- 
rections for  errors  committed  in  the  estimation  of  a  star's  place, 
while  parallax  serves  to  transfer  the  co-ordinates  from  the  earth's 
surface  to  its  centre.     The  object  of  the  reduction  of  observa- 
tions from  the  surface  to  the  centre  of  the  earth,  is  to  render  ob- 
servations made  at  different  places  on  the  earth's  surface  directly 
comparable  with  each   other.      Observers  occupying   different 
stations  upon  the  earth  refer  the  same  body,  unless  it  be  a  fixed 
star,  to  different  points  of  the  celestial  sphere.     Their  observa- 
tions cannot,  therefore,  be  compared  together,  unless  they  be  re- 
duced to  the  same  point,  and  the  centre  of  the  earth  is  the  most 
convenient  point  of  reference  that  can  be  chosen. 

REFRACTION. 

§O.  Atmospheric  Refraction.  We  learn  from  the  princi- 
ples of  Pneumatics,  as  well  as  by  experiments  with  the  barome- 
ter, that  the  atmosphere  gradually  decreases  in  density  from  the 
earth's  surface  upwards.  We  learn  also  from  the  same  sources, 
that  it  may  be  conceived  to  be  made  up  of  an  infinite  number 
of  strata  of  decreasing  density,  concentric  with  the  earth's  sur- 
face. From  the  known  pressure  and  density  of  the  atmosphere 
at  the  surface  of  the  earth,  it  is  computed,  that  by  the  laws  of 
the  equilibrium  of  fluids,  if  its  density  were  throughout  the 
same  as  immediately  in  contact  with  the  earth,  its  altitude  would 
be  about  5  miles.  Certain  facts,  hereafter  to  be  mentioned,  show 


REFRACTION. 


55 


that  its  actual  altitude  is  not  far  from  50  miles.  Now,  it  is 
an  established  principle  of  Optics,  that  light  in  passing  from  a 
vacuum  into  a  transparent  medium,  or  from  a  rarer  into  a  denser 
medium,  is  bent  or  refracted  towards  the  perpendicular  to  the 
surface  at  the  point  of  incidence.  It  follows,  therefore,  that  the 
light  which  comes  from  a  star,  in  passing  into  the  earth's  atmo- 
sphere, or  in  passing  from  one  stratum  of  atmosphere  into  another, 
is  refracted  towards  the  radius  drawn  from  the  centre  of  the 
earth  to  the  point  of  incidence. 

Path  of  a  ray  of  light.    Let  MmnN,  NnoO,  OoqQ,  (Fig.  29,) 


FIG.  29. 

represent  successive  strata  of  the  atmosphere.  Any  ray,  Sp,  will 
then,  instead  of  pursuing  a  straight  course,  Spa:,  follow  the 
broken  line  pabc ;  being  bent  downwards  at  the  points^?,  a,  5,  c, 
&c.,  where  it  enters  the  different  strata.  But,  since  the  number 
of  strata  is  infinite,  and  the  density  increases  by  infinitely  small 
degrees,  the  deflections  apx,  bay,  &c.,  as  well  as  the  lengths  of 
the  lines  pa,  ab,  &c.,  are  infinitely  small ;  and  therefore  pabc,  the 
path  of  the  ray,  is  a  broken  line  of  an  infinite  number  of  parts, 
or  a  curved  line  concave  towards  the  earth's  surface,  as  it  is  re- 
presented in  Fig.  30.  Moreover,  it  lies  in  the  vertical  plane  con- 
taining the  original  direction  of  the  ray ;  for  this  plane  is  per- 
pendicular to  all  the  strata  of  the  atmosphere,  and  therefore  the 
ray  will  continue  in  it  in  passing  from  one  to  the  other. 

81.  Astronomical  Refraction.  The  line  OS' (Fig.  30)  drawn 
tangent  to  paO,  the  curvilinear  path  of  the  light,  at  its  lowest 
point,  will  represent  the  direction  in  which  the  light  enters  the 
eye,  and  therefore  the  apparent  line  of  direction  of  the  star.  If, 
then,  OS  be  the  true  direction  of  the  star,  the  angle  SOS'  will 
be  the  displacement  of  the  star  produced  by  Atmospheric  Refrac- 
tion. This  angle  is  called  the  Astronomical  Refraction,  or  simply 
the  Refraction  of  the  star. 

Since  paO  is  concave  towards  the  earth,  OS'  will  lie  above 


56 


CORRECTIONS  OF  OBSERVATIONS. 


.OS ;  consequently,  refraction  makes  the  apparent  altitude  of  a  star 
greater  than  its  true  altitude,  and  the  apparent  zenith  distance  of  a 
star  less  than  its  true  zenith  distance.  (We  here  speak  of  the  true 
altitude  and  true  zenith  distance,  as  estimated  from  the  station 


ol  the  observer  upon  the  earth's  surface.)  Thus,  to  obtain  the 
true  altitude  from  the  apparent,  we  must  subtract  the  refraction; 
and  to  obtain  the  true  zenith  distance  from  the  apparent,  we  must 
add  the  refraction.  As  refraction  takes  effect  wholly  in  a  verti- 
cal plane  (80),  it  does  not  alter  the  azimuth  of  a  star. 

The  amount  of  the  refraction  varies  with  the  apparent  zenith  dis- 
tance. In.  the  zenith  it  is  zero,  since  the  light  passes  perpendi- 
cularly through  all  the  strata  of  the  atmosphere :  and  it  is  the 
greater,  the  greater  is  the  zenith  distance;  for,  the  greater  the 
zenith  distance  of  a  star,  the  more  obliquely  does  the  light  which 
comes  from  it  to  the  eye  penetrate  the  earth's  atmosphere,  and 
enter  its  different  strata,  and  therefore,  according  to  a  well-known 
principle  of  optics,  the  greater  is  the  refraction. 

82.  To  find  the  Amount  of  the  Refraction  for  a  given 
Zenith  Distance  or  Altitude.  Let  us  first  show  a  method  of 
resolving  this  problem  by  the  general  theo^  of  refraction.  Ac- 
cording to  this  theory,  the  amount  of  the  refraction,  except  so 
far  as  the  convexity  of  the  strata  of  the  atmosphere  may  have 
an  effect,  depends  wholly  upon  the  absolute  density  of  the  air 
immediately  in  contact  with  the  earth,  and  not  at  all  upon  the 
law  of  variation  of  the  density  of  the  different  strata ;  that  is,  the 
actual  refraction  is  the  same  that  would  take  place  if  the  light 
passed  from  a  vacuum  immediately  into  a  stratum  of  air  of  the 
density  which  obtains  at  the  earth's  surface.  Let  us  suppose,  then, 
that  the  whole  atmosphere  is  brought  to  the  same  density  as  that 
portion  of  it  which  is  in  contact  with  the  earth,  and  let  bah  (Fig. 
31)  represent  its  surface ;  also  let  O  represent  the  station  of  the 
observer  upon  the  earth's  surface,  and  Sa  a  ray  incident  upon  the 
atmosphere  at  a.  Denote  the  angle  of  refraction  OaC  by  p>  and 
the  refraction  Oax  by  r.  The  angle  of  incidence 


REFKACTION. 


57 


Z'aS  =  Z'oS'4-  S'aS  =  OaC  +  Oax  =p  +  r. 
Now  if  we  represent  the  index  of  refraction  of  the  atmo- 
sphere by  ra,  we  have,  by  a  law  of  refraction, 

sin  Z'aS  =  m  sin  OaC,  or  sin  (p  -f  r)  =  ra  sin  p; 


FIG.  31. 

developing,  (App.  For.  15,) 

sin  p  cos  r  +  cos^  sin  r  =  m  sin  p  ; 
or,  dividing  by  sin  p, 

cos  r  -f  cot  p  sin  r=m. 

But,  as  r  is  small,  we  may  take  cos  r  =  1,  and  sin  r  =  r  =  r" 
sin  1",  (App.  47.) 

Whence,  l+cotjp.r"sin  l"= 


putting  A  =      ~-     Let  ZCa  =  C  ;  and  ZOa  =  Z.     OaC  = 

sin  _L 

ZOa  —  ZCa,  or  jp  =  Z  —  C.  Substituting,  we  have  r"  =  A  tang 
(Z  —  C)  ;  or,  omitting  the  double  accent,  and  considering  r  as 
expressed  in  seconds, 

r  =  A  tang(Z  —  C)  .....  (2) 

When  the  zenith  distance  is  not  great,  C  is  quite  small  compared 
with  Z.  If  we  neglect  it,  we  have 

r  =  A  tang  Z  .....  (3)  ; 

which  is  the  expression  for  the  refraction,  answering  to  the  sup- 
position that  the  surface  of  the  earth  is  a  plane,  and  that  the 


58 


CORRECTIONS  OF  OBSERVATION'S. 


light  is  transmitted  through  a  stratum  of  uniformly  dense  air, 
parallel  to  its  surface.  We  perceive,  therefore,  that  the  refrac- 
tion, except  in  the  vicinity  of  the  horizon,  varies  nearly  as  the  tan- 
gent of  the  apparent  zenith  distance. 

It  has  been  ascertained  by  experiment  that  m,  the  index  of 
refraction  (the  barometer  being  =  29.6  inches,  and  the  ther- 
mometer =  50°),  =  1.0002803.  Substituting  in  equation  (3), 
after  having  restored  the  value  of  A,  and  reducing,  there  results 

r  =  57".8  tang  Z (4). 

83.  Formulae  of  Refraction.  With  the  aid  of  this  for- 
mula, or  of  others  purely  theoretical,  astronomers  have  sought 
to  determine  the  precise  amount  of  the  refraction  at  various 
zenith  distances  from  observation,  and  by  collating  the  results 
of  their  observations  to  obtain  empirical  formulae  that  are  more 
exact. 

One  of  the  simplest  methods  of  accomplishing  this  is  the  following :  When  the 
latitude  or  co-latitude  of  a  place,  and  the  polar  distance  of  a  star  which  passes  the 
meridian  near  the  zenith,  have  been  determined,  the  refraction  may  be  found 

for  all  altitudes  from  observation  simply. 
For,  let  P  (Fig.  32)  be  the  elevated  pole,  Z 
the  zenith,  PZE  the  meridian,  HOB  the 
horizon,  S  the  true  place  of  a  star,  and  S' 
its  apparent  place.  Suppose  the  apparent 
zenith  distance  ZS'  to  have  been  mea- 
sured. Now,  in  the  triangle  ZPS,  ZP 
the  co-latitude  and  PS  the  polar  distance 
are  known  by  hypothesis,  and  the  angle 
P  is  the  sidereal  time  which  has  elapsed 
since  the  star's  last  meridian  transit,  (or, 
if  the  star  be  to  the  east  of  the  meridian, 
the  difference  between  this  interval  and 
24  sidereal  hours,)  converted  into  degrees 
by  allowing  15°  to  the  hour.  Therefore 
we  may  compute  the  true  zenith  distance 
ZS,  and  subtracting  from  it  the  apparent 
zenith  distance  ZS',  we  shall  have  the  re- 
fraction. For  the  solution  of  this  prob- 
lem, the  polar  distance  may  be  found  by 
taking  the  complement  of  the  declination  computed  from  an  observed  meridian 
zenith  distance  (55) ;  and,  since  the  upper  and  lower  transits  of  a  circumpolar  star 
take  place  at  equal  distances  from  the  pole,  the  co-latitude  may  be  found  by  tak- 
ing the  half  sum  of  the  greatest  and  least  zenith  distances  of  the  pole-star.  But 
it  is  obvious  that  neither  of  these  quantities  can  be  accurately  determined,  unless 
the  measured  zenith  distances  be  corrected  for  refraction.  When,  however,  the 
zenith  distances  in  question  differ  considerably  from  90°,  the  corresponding  refrac- 
tions may  be  at  first  ascertained  with  considerable  accuracy  by  means  of  equation 
(4).  When  more  correct  formulas  have  been  obtained  by  this  or  any  other  pro- 
cess, the  latitude  and  polar  distance,  and  therefore  the  refraction  answering  to  the 
measured  zenith  distance,  will  become  more  accurately  known. 

The  various  formulae  of  refraction  having  been  tested  by  nu- 
merous observations,  it  is  found  that  they  are  all,  though  in  dif- 
ferent degrees,  liable  to  material  errors  when  the  zenith  distance 
exceeds  80°,  or  thereabouts.  At  greater  zenith  distances  than 


FIG.  32. 


REFRACTION.  59 

this  the  refraction  is  irregular,  or  is  frequently  different  in  amount 
when  the  circumstances  on  which  it  is  supposed  to  depend  are 
the  same. 

§4.  Mean  Refractions. —  Corrections  for  the  varying 
density  of  the  Air.  The  refractive  power  of  the  air  varies 
with  its  density,  and  hence  the  refractions  must  vary  with  the 
height  of  the  barometer  and  thermometer.  The  refractions 
which  have  place  when  the  barometer  stands  at  30  inches  and 
the  thermometer  at  50°,  are  called  mean  refractions.  The  refrac- 
tions corresponding  to  any  other  height  of  the  barometer  or 
thermometer,  are  obtained  by  seeking  the  requisite  corrections  to 
be  applied  to  the  mean  refractions  in  consequence  of  the  differ- 
ence between  the  actual  density  of  the  air  and  its  assumed  mean 
density. 

Tables  of  Refraction.  To  save  astronomical  observers  the 
trouble  of  calculating  the  refraction  whenever  it  is  needed,  the 
mean  refractions  corresponding  to  various  zenith  distances,  or  al- 
titudes, are  computed  from  the  formula,  as  also  the  corrections 
for  various  heights  of  the  barometer  and  thermometer,  and  in- 
serted in  tables.  (See  Tables  VIII.  and  IX.) 

On  inspecting  Table  VIII.,  it  will  be  seen  that  the  refraction 
amounts  to  about  34'  when  a  body  is  in  the  apparent  horizon, 
and  to  about  58"  when  it  has  an  altitude  of  45°. 

§5.  Other  Effects  of  Atmospheric  Refraction.  Atmo- 
spheric refraction  makes  the  apparent  distance  of  any  two  heav- 
enly bodies  less  than  the  true ;  for  it  elevates  them  in  vertical 
circles  which  continually  approach  each  other  from  the  horizon 
till  they  meet  in  the  zenith. 

Kefraction  also  gives  to  the  discs  of  the  sun  and  moon  an 
eliptical  form  when  near  the  horizon,  As  it  increases  with  an 
increase  of  zenith  distance,  the  lower  limb  of  the  sun  or  moon 
is  more  refracted  than  the  upper,  and  thus  the  vertical  diameter 
is  shortened,  while  the  horizontal  diameter  remains  the  same,  or 
very  nearly  so.  This  effect  is  greatest  near  the  horizon,  for  the 
reason  that  the  increase  of  the  refraction  is  there  the  most  rapid ; 
and  it  is  most  observable  at  sea,  as  the  sun  and  moon,  at  their 
rising  or  setting,  can  there  be  seen  in  closer  proximity  to  the 
horizon  than  at  most  stations  on  land.  The  difference  between 
the  vertical  and  horizontal  diameters  may  amount  to  I  part  of 
the  whole  diameter. 

When  a  star  appears  to  be  in  the  horizon,  it  is  actually  34' 
below  it  (84) :  retraction,  then,  retards  the  setting  and  accele- 
rates the  rising  of  the  heavenly  bodies. 

Having  this  effect  upon  the  rising  and  setting  of  the  sun,  it 
must  increase  the  length  of  the  day. 

The  apparent  diameter  of  the  sun  is  about  32' ;  as  this  is  less 
than  the  refraction  in  the  horizon,  it  follows,  that  when  the  sun 
appears  to  touch  the  horizon  it  is  actually  entirely  below  it.  The 


CORRECTIONS  OF  OBSERVATIONS. 


same  is  true  of  the  moon,  as  its  apparent  diameter  is  nearly  the 
same  with  that  of  the  sun. 


S6.    Definition*. 


PARALLAX.  j. 

The  correction  for  atmospheric  refraction  having  been  ap- 
plied, the  zenith  distance  of  a  body  is  reduced  from  the  surface 
of  the  earth  to  its  centre,  by  means  of  a  correction  called  Par- 
allax. 

Parallax  is,  in  its  most  general  sense,  the 
angle  made  by  the  lines  of  direction,  or 
the  arc  of  the  celestial  sphere  comprised 
between  the  places  of  an  object,  as  viewed 
from  two  different  stations.  It  may  also 
be  defined  to  be  the  angle  subtended  at 
an  object  by  a  line  joining  two  different 
places  of  observation.  Let  S  (Fig.  33)  re- 
present a  celestial  object,  and  A  B  two 
places  from  which  it  is  viewed.  At  A  it 
will  be  referred  to  the  point  s  of  the  ce- 
lestial sphere,  and  at  B-to  the  point  s' ; 
the  angle  BSA,  or  the  arc  ss',  is  the  paral- 
lax. The  arc  ss'  is  taken  as  the  measure  of 
the  angle  BSA,  on  the  principle  that  the 
celestial  sphere  is  a  sphere  of  an  indefi- 
nitely great  radius,  so  that  the  point  S 
is  not  sensibly  removed  from  its  centre. 
The  term  parallax  is,  however,  generally  used  in  Astronomy 
in  a  limited  sense  only,  namely,  to  denote  the  angle  included  be- 
tween the  lines  of  direction  of  a  heavenly  body,  as  seen  from  a 
point  on  the  earth's  surface  and  from  its  centre  ;  or  the  angle  sub- 
tended at  a  heavenly  body  by  a  radius  of  the  earth.  If  C  (Fig. 
34)  is  the  centre  of  the  earth,  0  a  point  on  its  surface,  and  S  a 
heavenly  body,  OSC  is  the  parallax  of  the  body.  When  there 
is  occasion  to  distinguish  this  angle  from  other  angles  of  paral- 
lax, it  is  termed  the  Geocentric  Parallax. 

The  parallax  of  a  heavenly  body  above  the  horizon  is  called 
Parallax  in  Altitude. 

The  parallax  of  a  body  at  the  time  its  apparent  altitude  is 
zero,  or  when  it  is  in  the  plane  of  the  horizon,  is  called  the  Hori- 
zontal Parallax  of  the  body.  Thus,  if  the  body  S  (Fig.  34)  be 
supposed  to  cross  the  plane  of  the  horizon  at  S',  OS'C  will  be 
its  horizontal  parallax.  OSC  is  a  parallax  in  altitude  of  this 
body. 

It  is  to  be  observed,  that  the  definition  just  given  of  the  hori- 
zontal parallax,  answers  to  the  supposition  that  the  earth  is  of  a 
spherical  form.  In  point  of  fact,  the  earth  (as  will  be  shown  in 


PARALLAX. 


61 


the  sequel)  is  a  spheroid,  and  accordingly  the  vertical  and  the 
radius  at  any  point  of  its  surface  are  inclined  to  each  other ;  as 


FIG.  34. 


represented  in  Fig.  35,  where  00  is  the  radius,  and  00'  the  ver- 
tical.   The  points  Z  and  z,  in  which  the  vertical  and  radius 


FIG.  35. 


pierce  the  celestial  sphere,  are  called,  respectively,  the  Apparent 
Zenith  and  the  True,  or  Central  Zenith.  In  perfect  strictness,  the 
horizontal  parallax  is  the  parallax  at  the  time  sOS,  the  apparent 


62  CORRECTIONS  OF  OBSERVATIONS. 

distance  from  the  true  zenith,  is  90°.  But  no  material  error 
will  be  committed  in  supposing  the  earth  to  be  spherical,  except 
the  question  relates  to  the  parallax  of  the  moon. 

87.  True  Zenith  Distance.  Let  the  apparent  zenith  dis- 
tance  ZOS  =  Z,  (Fig.  34,)  the  true  zenith  distance  ZCS  =  z, 
and  the  parallax  OSC  =  p.  Since  the  angle  ZOS  is  the  ex- 
terior angle  of  the  triangle  OSC,  we  have 

ZOS  =  ZCS  +  OSC,  and  hence  also  ZCS  =  ZOS  —  OSC ; 
or, 

Z  =  z  +  p,  and  z  =  Z  —  p  .  .  .  .  (5). 

Thus,  to  obtain  the  true  zenith  distance  from  the  apparent,  we, 
have  to  subtract  the  parallax  ;  and  to  obtain  the  apparent  zenith 
distance  from  the  true,  to  add  the  parallax. 

Parallax,  then,  takes  effect  wholly  in  a  vertical  plane,  like  the 
refraction,  but  in  the  inverse  manner;  depressing  the  star,  while 
the  refraction  elevates  it.  Thus,  the  refraction  is  added  to  Z,  but 
tjfe  parallax  is  subtracted  from  it. 

§§.  To  find  an  Expression  for  the  Parallax  in  Alti- 
tude, in  terms  of  the  apparent  zenith  distance.  In  the  triangle 
SOC  (Fig.  34)  the  angle  OSC  =  parallax  in  altitude  =^  OC  = 
radius  of  the  earth  =  R,  CS  =  distance  of  the  body  S  =  D,  and 
COS  =  180°—  ZOS  =  180°—  apparent  zenith  distance  =  180° 
—  Z  ;  and  we  have  by  Trigonometry  the  proportion 

sin  OSC  :  sin  COS  ::  CO  :  CS ; 
whence, 

sin^? :  sin  (180°—  Z)  ::  K  :  D ; 

and 

D  sin  p  =»  R  sin  Z ; 

B 

sin  p  —  — -  sin  Z (6). 

This  equation  shows  that  the  parallax  p  depends  for  any  given 
zenith  distance  Z  upon  the  distance  of  the  body,  and  is  less  in 
proportion  as  this  distance  is  greater:  also,  that  for  any  given 
distance  of  the  body  it  increases  with  an  increase  in  the  zenith 
distance.  When  Z  =  90°,  p  has  its  maximum  value,  and  then 
=  horizontal  parallax  =  H ;  and  equa.  (6)  gives 

sinH=g. (7); 

substituting,  we  have 

smp  =  sin  H  sin  Z  ....  (8). 

This  equation  may  be  somewhat  simplified.  The  distances 
of  the  heavenly  bodies  are  so  great,  that  p  and  H  are  always 
very  small  angles ;  even  for  the  moon,  which  is  much  the  near- 
est, the  value  of  H  does  not  at  any  time  exceed  62'.  We  may, 


PARALLAX.  63 

therefore,  without  material  error,  replace  sin  p  and  sin  H  by  p 
und  H.     This  being  done,  there  results, 

p  =  H  sin  Z  ____  (9). 

Wherefore,  the  parallax  in  altitude  equals  the  product  of  the  hori- 
zontal parallax  by  the  sine  of  the  apparent  zenith  distance. 

If  we  take  notice  of  the  deviation  of  the  earth's  form  from 
that  of  a  sphere,  Z,  in  equation  (8),  will  represent  the  apparent 
distance  from  the  true  zenith,  (86,)  and  H  the  horizontal  paral- 
lax as  it  is  defined  in  Art.  86. 

In  order  to  be  able  to  compute  the  parallax  in  altitude  by 
means  of  formula  (9)  it  is  necessary  to  know  H,  the  horizontal 
parallax. 

§9.  To  find  an  Expression  for  tbe  Horizontal  Paral- 
lax, in  terms  of  measurable  quantities.  Let  O,  O'  (Fig.  35)  repre- 
sent two  stations  upon  the  same  terrestrial  meridian  OEO',  and 
remote  from  each  other,  Z,  Z7  their  apparent  zeniths,  and  z,  z'  their 
true  zeniths,  QCE  the  equator,  and.S  the  body  (supposed  to  be 
in  the  meridian)  the  parallax  of  which  is  to  be  found.  Let  the 
angle  OSO'  =  A,  zOS  =  Z,  z'O'S  =  Z7;  also  let  CO  =  R,  CO' 
—  R',  CS  =  D,  the  parallax  in  altitude  OSC  =  p,  and  the  par- 
allax in  altitude  O'SC  =  p'.  Now,  by  equation  (6),  replacing 
the  sine  of  the  parallax  by  the  parallax  itself,  (88,) 

p  =  —  sin  Z,  and  pf  =.  j*'  sin  Z'; 

whence 

R    ,  R'  .  RsinZ-hR'sinZ' 

P  +  y=j)-smZ+j5-sinZ/=  -p-      -; 

and,  (equ.  7,) 

R  R 


Substituting  this  value  of  D,  and  deducing  the  value  of  H,  we 
have 


H 


R  (p  +  /)         ___  R  X  A  _ 
RsinZ  H-  R'sinZ'~~RsinZ  +  R'  sin  Z'  ' 


It  remains  now  to  find  an  expression  for  A  in  terms  of  mea- 
surable quantities.  Let  Os  and  O's  (Fig.  35)  be  the  directions,  at 
O  and  O',  of  a  fixed  star  which  crosses  the  meridian  nearly  at  the 
same  time  with  the  body.  Owing  to  the  immense  distance  of  the 
star,  these  lines  will  be  sensibly  parallel  to  each  other  (19).  Let 
the  angle  SOs,  the  difference  between  the  meridian  zenith  dis- 
tances of  the  body  and  star,  as  observed  at  O,  be  represented  by 
d,  and  let  the  same  difference  SO's  for  the  station  O',  be  repre- 
sented by  d  '.  Now, 

OSO'  =  OLO'  —  SO's  =  80s—  SO'*,  or  A  =  d—  d'. 


64  CORRECTIONS  OF  OBSERVATIONS. 

If  the  body  be  seen  on  different  sides  of  the  star  by  the  two 
observers,  we  shall  have 


Substituting  in  equation  (11),  there  results, 

H-        E  (<*±<y)  (12) 

RsinZ+R'sinZ'  ' 

If  we  regard  the  earth  as  a  sphere,  R=R',  and  dividing  by 
R,  we  have 

TT  _ 

~ 


s  mz' 

90.  To    Determine    the    Horizontal    Parallax   of    a 
body,  from  Observation  ;  by  means  of  this  formula.     Let  each 
of  the  two  observers  measure  the  meridian  zenith  distance  of  the 
body,  and  also  of  a  star  which  crosses  the  meridian  nearly  at 
the  same  time  with  the  body,  and  correct  the  measured  distances 
for  refraction.     The  difference  of  the  two  will  be,  respectively, 
the  value  of  d  and  d';  and  the  corrected  zenith  distances  of  the 
body  will  be  the  values  of  Z  and  Z'.     If  formula  (12)  be  used, 
the  measured  zenith  distances  of  the  body  must  still  be  corrected 
for  the  reduction  of  latitude,  (Art.  23,  def.  4.) 

It  is  not  necessary  that  the  two  stations  should  be  on  precisely 
the  same  meridian  ;  for  if  the  meridian  zenith  distance  of  the 
body  be  observed  from  day  to  day,  its  daily  variation  will  become 
known  ;  then,  knowing  also  the  difference  of  longitude  of  the 
two  places,  the  following  simple  proportion  will  give  the  change 
of  zenith  distance  during  the  interval  of  time  employed  by  the 
body  in  moving  from  the  meridian  of  the  most  easterly  to  that 
of  the  most  westerly  station,  viz.  :  as  interval  (T)  of  two  suc- 
cessive transits  :  diff.  of  long.,  expressed  in  time,  (t)  ::  varia- 
tion of  zenith  dist.  in  interval  T  :  its  variation  in  interval  t. 
This  result,  applied  to  the  zenith  distance  observed  at  one  of  the 
stations,  will  reduce  it  to  what  it  would  have  been  if  the  obser- 
vation had  been  made  in  the  same  latitude  on  the  meridian  of  the 
other  station. 

The  horizontal  parallax  of  the  moon  has  been  determined  by 
this  process  with  sufficient  accuracy.  The  parallaxes  of  the  sun 
and  planets,  which  are  very  small,  have  been  determined  by 
much  more  accurate  methods.  The  importance  of  having 
recourse  to  methods  of  the  greatest  possible  accuracy,  in  the  case 
of  the  sun  and  planets,  will  appear  in  the  sequel. 

91.  Horizontal  Parallax  in  Different   Latitude**.     In 
consequence  of  the  spheroidal  form  of  the  earth,  the  horizontal 
parallax  of  a  body  is  somewhat  different  in  different  latitudes. 
Let  H  and  H'  denote  the  horizontal  parallaxes  of  the  same  body, 
at  the  distance  D,  and  R  and  R'  the  radii  of  the  earth  at  two 
different  latitudes  ;  then,  by  equ.  7, 


PARALLAX.  65 

sin  H  =  ?,  and  sin  K'=?L ; 

7?       T?' 
whence,  sin  H  :  sin  H'::  jj  •  -p  ::  R  :  K'. 

Also,  as  H  and  H'  are  small,  we  have  very  nearly, 
H:H'::K:K'. 

Thus  the  horizontal  parallax  is  greatest  at  the  equator,  and 
decreases  nearly  in  the  same  ratio  with  the  radius  of  the  earth 
from  the  equator  to  the  poles.  The  horizontal  parallax  of  the 
moon  is  about  11"  greater  at  the  equator  than  at  the  poles.  In 
the  case  of  the  sun,  or  of  any  planet,  the  difference  is  in  every 
instance  less  than  £". 

9*2.  Equatorial  Parallax.  The  horizontal  parallax  of  a 
body,  for  a  station  on  the  equator,  is  called  its  equatorial  hori- 
zontal parallax,  or  simply  its  equatorial  parallax. 

The  equatorial  parallax  of  the  moon  varies  from  52'  50"  to 
61'  32",  according  to  the  distance  of  the  moon  from  the  earth. 
At  the  mean  distance  its  value  is  57'  3". 

The  equatorial  horizontal  parallax  of  the  sun,  at  the  earth's 
mean  distance,  is  8".95.  The  sun's  horizontal  parallax  varies 
with  the  earth's  distance  less  than  -J". 

The  horizontal  parallaxes  of  the  planets,  at  their  varying  dis- 
tances from  the  earth,  are  comprised  between  the  limits  34"  and 
0/'3.  The  greater  limit  is  the  parallax  of  Yenus  when  nearest 
the  earth,  and  the  smaller  limit  is  the  parallax  of  Neptune  when 
farthest  from  the  earth. 

The  fixed  stars  have  no  geocentric  parallax. 

Tables  of  Parallax.  In  the  present  condition  of  astronomical 
science,  the  horizontal  parallax  of  the  sun,  moon,  or  any  planet, 
may  be  calculated  for  any  particular  time  from  the  results  of 
astronomical  observations,  or  may  more  readily  be  obtained  by 
the  aid  of  tables  that  have  been'  computed  for  the  purpose  of 
facilitating  its  determination.  It  may  also  be  obtained  by  simple 
inspection,  from  the  Nautical  Almanac.  The  American,  or  Eng- 
lish Nautical  Almanac,  is  a  collection  of  data  to  be  used  in 
nautical  and  astronomical  calculations,  published  annually,  two 
or  three  years  in  advance  of  the  year  for  which  it  is  calculated. 

93.  Parallax  in  Right  Ascension  and  in  Declination. 
Since  the  parallax  of  a  body  displaces  it  in  its  vertical  circle, 
which  is  generally  oblique  to  the  equator,  it  will  alter  its  right 
ascension  and  declination.  The  consequent  corrections  to  be 
applied  to  the  right  ascension  and  declination  are  called,  respec- 
tively, parallax  in  right  ascension,  and  parallax  in  declination. 

For  a  similar  reason  the  parallax  of  a  body,  generally  alters 
both  its  longitude  and  latitude ;  and  the  requisite  corrections  are 
termed  parallax  in  longitude,  and  parallax  in  latitude. 

5 


CORRECTIONS  OF  OBSERVATIONS. 


Formulae  for  calculating  the  parallax  in  right  ascension,  and 
in  declination,  as  well  as  in  longitude  and  latitude,  are  investi- 
gated in  the  Appendix. 


ABERRATION. 

91.  The  celebrated  English  astronomer,  Dr.  Bradley,  com- 
menced in  'the  year  1725  a  series  of  accurate  observations, 
with  the  view  of  ascertaining  whether  the  apparent  places 
of  the  fixed  stars  were  subject  to  any  direct  alteration  in 
consequence  of  the  continual  change  occurring  in  the  earth's 
position  in  space.  The  observations  showed  that  there  had  been 
in  reality,  during  the  period  of  observation,  small  changes  in  the 
apparent  places  of  each  of  the  stars  observed,  which,  when 
greatest,  amounted  to  about  40" ;  but  they  were  not  such  as 
should  have  resulted  from  the  orbital  motion  of  the  earth.  These 
phenomena  Dr.  Bradley  undertook  to  examine  and  reduce  to  a 
general  law.  After  repeated  trials,  he  at  last  succeeded  in  dis- 
covering their  true  explanation.  His  theory  is,  that  they  are 
different  effects  of  one  general  cause,  a  progressive  motion  of 
light  in  conjunction  with  the  orbital  motion  of  the  earth. 

95.  Aberration  of  Light.  Let  us  conceive  the  observer 
to  be  stationed  at  the  earth's  centre ;  and  let  ACB  (Fig.  36)  be  a 


portion  of  the  earth's  orbit,  so  small  that  it  may  be  considered  a 
right  line,  OS  the  true  direction  of  a  fixed  star  as  seen  from  the  point 
C,  AC  the  distance  through  which  the  earth  moves  in  some  small 
portion  of  time,  and  aC  the  distance  traversed  by  a  wave  of  light, 
in  the  same  time.  Then,  a  ray  of  light,  which,  coming  from  the 
star  in  the  direction  SO,  is  at  a  at  the  same  time  that  the  earth  is 
at  A,  will  arrive  at  C  at  the  same  time  that  the  earth  does.  Sup- 
pose that  Aa  is  the  position  of  the  axis  or  central  line  of  a  tele- 


ABERRATION.  67 

scope,  wheii  the  earth  is  at  A,  and  that,  continuing  parallel  tc 
itself,  it  takes  up,  by  virtue  of  the  earth's  motion,  the  successive 

positions  A V,  A" a" CS'.  A  ray  of  light  which  follows 

the  line  SC  in  space  will  descend  along  this  axis:  for  aa!  is  to 
A  A'  and  CM"  is  to  A  A",  as  aC  is  to  AC,  that  is,  as  the  velocity 
of  light  is  to  the  velocity  of  the  earth  ;  consequently,  when  the 
earth  is  at  A7  the  ray  of  light  is  on  the  axis  at  a',  and  when  the 
earth  is  at  A"  the  ray  is  on  the  axis  at  a",  and  so  on  for  all  the 
other  positions  of  the  axis,  until  the  earth  arrives  at  C.  The 
apparent  direction  of  the  star  S,  as  far,  at  least,  as  it  depends 
upon  the  cause  under  consideration,  will  therefore  be  CS'. 

The  angle  SCS',  which  expresses  the  change  in  the  apparent 
place  of  a  star  S,  produced  by  the  motion  of  light  combined  with 
the  motion  of  the  spectator,  is  called  the  Aberration  of  the  star  ; 
and  the  phenomenon  of  the  change  of  the  apparent  course  of  the 
light  coming  from  a  star,  thus  produced,  is  called  Aberration  of 
Light,  or  simply  Aberration. 

The  phenomenon  of  the  aberration  of  light  may  be  familiarly 
illustrated  by  taking  falling  drops  of  rain  instead  of  supposed  par- 
ticles of  light,  and  a  vessel  in  motion  at  sea  instead  of  the  earth 
moving  through  space ;  and  considering  what  direction  must  be 
given  to  a  small  tube  by  a  person  standing  upon  the  deck  of  the 
vessel,  so  as  to  permit  the  drops  falling  perpendicularly  to  pass 
through  the  tube.  It  is  plain  that  if  the  tube  had  a  precisely  verti- 
cal position,  its  forward  motion  would  bring  the  back  part  of  the 
tube  against  the  drop ;  and  that  the  only  way  to  prevent  this  is 
to  incline  the  upper  end  of  the  tube  forward,  or  draw  the  lower 
end  backward,  whereby  the  back  part  of  it  would  be  made  to 
pass  through  a  greater  distance  before  it  comes  up  to  the  line  of 
descent  of  the  drop.  The  quantity  that  it  is  made  to  deviate  in 
direction  from  this  line,  must  depend  upon  the  relative  velocities 
of  the  falling  drop  and  moving  tube.  To  the  observer,  uncon- 
scious of  his  own  motion,  the  drop  will  appear  to  fall  in  the 
oblique  direction  of  the  tube. 

96.   Angle  of  Aberration.  If  through  the  point  a  (Fig.  37) 


FIG.  37. 


a  line,  as',  be  drawn  parallel  to  AC,  and  terminating  in  CS',  the 
figure  Aos'C  will  be  a  parallelogram,  and  therefore  as'  will  be 
equal  to  AC.  Hence  it  appears,  that  if  on  CS,  the  line  of  direc- 


ov^ 

„     ••* 

68  CORRECTIONS  OF  OBSERVATIONS. 

tion  of  a  star  S,  a  line  Ca  be  laid  off,  representing  the  velocity 
of  light,  and  through  a  a  line,  as',  be  drawn,  having  the  same 
direction  as  the  earth's  motion  and  equal  to  its  velocity,  the  line 
joining  s'  and  C  will  be  the  apparent  line  of  direction  of  the  star, 
the  point  S'  its  apparent  place  in  the  heavens,  and  the  angle  aCs' 
its  aberration.  We  conclude,  therefore,  that  by  virtue  of  aber- 
ration a  star  is  seen  in  advance  of  its  true  place,  in  the  plane 
passing  through  the  line  of  direction  of  the  star  and  the  line  of 
the  earth's  motion. 

The  amount  of  the  aberration  of  a  star  is  always  very  small 
(never  greater  than  about  20"),  because  of  the  very  great  dispro- 
portion between  the  velocity  of  light  and  the  velocity  of  the 
earth.  It  is  very  much  exaggerated  in  Figs.  36  and  37. 

The  aberration  is  the  same  when  a  star  is  viewed  with  the  naked 
eye  as  when  it  is  seen  through  a  telescope.  For,  let  aC,  the 
velocity  of  the  light,  be  decomposed  into  two  velocities,  of  which 
one,  AC,  is  equal  and  parallel  to  the  velocity  of  the  earth,  the 
other  will  be  represented  by  s'C.  Now,  since  the  velocity  AC  is 
equal  and  parallel  to  the  velocity  of  the  earth,  it  will  produce  no 
change  in  the  relative  position  of  a  supposed  particle  of  light  and 
the  eye,  and  therefore  the  relative  motion  of  the  light  and  the 
eye  will  be  the  same  that  it  would  be  if  the  earth  were  stationary 
and  the  light  had  only  the  velocity  s'C  ;  accordingly,  the  light 
entering  the  eye  just  as  it  would  do  if  it  actually  came  in  the 
direction  s'C,  and  the  eye  were  at  rest,  Cs'  will  be  the  apparent 
direction  of  the  star  from  which  it  proceeds. 

If  we  regard  the  observer  as  situated  upon  the  earth's  surface, 
instead  of  being  at  its  centre,  the  aberration  resulting  from  the 
earth's  motion  of  revolution  will  be  still  the  same,  for  all  points 
of  the  earth  advance  at  the  same  rate  and  in  the  same  direction 
with  the  centre.  The  motion  of  rotation  will  produce  an  aberra- 
tion proper  to  itself,  but  it  is  so  small  that  there  is  no  occasion 
to  take  it  into  account. 

*     97.  To  find  a  General  Expression  for  the  Aberration. 
We.  have  by  Trigonometry  (Fig.  37), 

sin  AaC  :  sin  CAa  :  :  CA  :  Ca  :  :  vel.  of  earth  :  vel.  of  light  ; 
whence, 


sin  AaC  =  sin  CAa  ^±  ;  or,  since  AaC  =  SCS', 
Ca 

sin  aberr.  =  sin  CAa  veL  ofeartl1.  .  .  .(14). 
vel.  of  light 

When  CAa  is  90°,  the  aberration  has  its  maximum  value,  and 
this  has  been  found  by  observation  to  be  20".445  ;  whence, 

sin  20".  445  =  ™1.  of  earth   .  .  .(15)  : 
vel.  of  light 


ABERRATION.  69 

substituting,  and  taking  sin  BCa  for  sin  CAa,  to  which  it  is  very 
nearly  equal,  we  have 

sin  aberr.  =  sin  BCa  sin  20".4±5. .  .  .(16). 

We  may  conclude  from  this  equation,  that  the  aberration  in- 
creases with  the  angle  BCa  made  by  the  direction  of  the  star 
with  the  direction  of  the  earth's  motion  ;  that  it  is  equal  to  zero 
when  this  angle  is  zero,  and  has  its  maximum  value  of  20//.445 
when  this  angle  is  90°. 

98.  A BS u u ul  Curve  of  Aberration.  Let  us  now  inquire  into 
the  entire  effect  of  aberration  in  the  course  of  a  year.  Let  S 
(Fig.  38)  be  the  sun ;  E  the  earth  ;  E#  its  orbit ;  ZTV  that  orbit 


FIG.  38. 

extended  to  the  fixed  stars,  or  the  ecliptic  (p.  15,  def.  17) ;  ET 
a  tangent  to  the  earth's  orbit  at  E  ;  ©  the  place  of  S  among  the 
fixed  stars  or  in  the  ecliptic,  as  seen  from  the  earth  ;  s  a  fixed 
star ;  sZT  the  arc  of  a  great  circle  passing  through  s  and  T. 
Then,  by  what  has  preceded  (96),  the  earth  moving  in  the  direc- 
tion E/J',  the  apparent  place  of  the  star  rnav  be  represented  by  s' 
and  the  aberration  by  sEs'.  Thus,  the  effect  of  aberration  at  any 
one  time  is  to  displace  the  star  by  a  small  amount,  directly 
towards  the  point  T  of  the  ecliptic,  which  is  90°  behind  the  sun. 
As  the  earth  moves,  the  position  of  the  point  T  will  vary ;  and 
in  the  course  of  a  year,  while  the  earth  describes  its  entire  orbit 
in  the  direction  E$r,  this  point  will  move  in  the  same  direction 
entirely  around  the  ecliptic.  In  this  period  of  time,  therefore, 
ss',  the  small  arc  of  aberration,  will  revolve  entirely  around  s,  the 
true  position  of  the  star ;  from  which  we  conclude,  that  in  conse- 
quence of  aberration  a  star  appears  to  describe  a  closed  curve  in 
the  heavens  around  its  true  place. 

As  the  inclination  of  the  direction  of  the  star  to  the  direction 
of  the  earth's  motion  will  vary  during  a  revolution  of  the  earth, 
the  aberration  will  also  vary  during  this  period  (97),  and  hence 
the  curve  in  question  will  not  be  a  circle.  It  appears  upon  inves- 
tigation that  it  is  an  ellipse,  having  the  true  place  of  tne  star  for 


70  CORRECTIONS  OF   OBSERVATIONS. 

its  centre,  and  of  which  the  semi-major  axis  is  constant  and  equal 
to  20".445,  and  the  semi-minor  axis  variable  and  expressed  by 
20//.445  sin  /L  (JL  denoting  the  latitude  of  the  star).  Each  star, 
then,  describes  an  ellipse  which  is  the  more  eccentric  in  propor- 
tion as  the  star  is  nearer  to  the  ecliptic;  for,  the  expression 
for  the  minor  axis  shows  that  the  smaller  the  latitude  the  less 
will  be  this  axis.  For  a  star  situated  in  the  ecliptic  the  minor 
axis  will  be  zero,  and  the  ellipse  will  be  reduced  to  a  right  line. 
For  a  star  in  the  pole  of  the  ecliptic  the  minor  axis  will  be  equal 
to  the  major,  and  the  ellipse  therefore  becomes  a  circle. 

In  following  the  motion  of  the  star  in  its  ellipse,  it  is  to  be  observed  that  the 
orbit  of  the  earth  is  a  mere  point  at  the  centre  of  the  celestial  sphere,  and  the 
angle  sET  as  the  earth  moves  forward,  decreases  from  90°  at  E  to  its  minimum 
value  at/,  and  then  increases  to  90°  at  r;  and  that  similar  changes  occur  while 
the  earth  is  describing  the  other  half  of  its  orbit.  When  the  earth  is  at  E,  the 
star  is  at  one  extremity  of  the  major  axis  of  its  ellipse,  and  when  the  earth  is 
arrived  at  r,  the  star  is  at  the  opposite  extremity  of  the  major  axis.  The  points  j 
and  g,  where  the  angle  sET  has  its  minimum  value,  answer  to  the  extremities  of 
the  minor  axis. 

99.  Aberration  of  the  sun. — Displacement  of  the  moon  and  planets.  Since  the 
motion  of  the  earth  is  at  all  times  in  a  direction  perpendicular,  or  nearly  so,  to  the 
line  followed  by  the  light  which  comes  from  the  sun  to  the  earth,  the  aberration 
of  the  sun,  which  takes  place  only  in  longitude,  is  continually  equal  to  about 
20".44.  Thus  the  sun's  apparent  place  is  always  about  20".44  behind  its  true 
place. 

The  apparent  displacement  of  a  planet,  resulting  from  the  progressive  motion  of 
light,  differs  from  that  of  a  fixed  star  in  a  similar  position.  As  a  planet  changes 
its  place  during  the  interval  of  time  that  a  ray  of  light  is  passing  from  it  to  the 
earth,  it  would,  if  the  earth  were  stationary,  appear  to  be  as  far  behind  its  true 
place  as  it  has  moved  during  this  interval.  This  angular  displacement,  dependent 
upon  the  motion  of  the  planet,  combined  with  the  aberration  proper  due  to  the 
earth's  motion,  constitutes  the  actual  angular  displacement  of  the  planet  from  the 
cause  under  consideration. 

The  apparent  change  of  place  caused  by  the  motion  of  the  moon  around  the 
earth  is  very  small. 

100.  Aberration  in  Right  A§cen§ioii  and  in  Decliiia- 
tioii.     Since  aberration  causes  the  apparent  place  of  a  star,  that 
has  been  corrected  for  refraction,  to  differ  slightly  from  its  true 
place,  the  true  and  apparent  co-ordinates  will  differ  somewhat 
from  each  other.     The  effects  of  the  aberration  of  light  upon  the 
right  ascension  and  declination  of  a  star  are  called,  respectively, 
the  aberration  in  right  ascension  and  the  aberration  in  declination. 
These  are  to  be  determined  and  applied  as  corrections  to  the 
apparent  right  ascension  and  declination ;  the  result  will  be  the 
true  co-ordinates,  which  will  define  the  actual  place  in  the  heavens 
of  the  body  observed. 

Formulae  for  computing  the  aberrations  of  a  star  in  right  as- 
cension and  declination,  are  investigated  in  the  Appendix. 

101.  Proof  of  the  Progressive  Motion  of  Light.     If  the 
apparent  places  of  a  star,  found  at  various  times,  be  corrected  for 
aberration,  the  same  result  for  the  true  place  of  the  star  is 
obtained.     Again,  the  deductions  of  Art.  98  agree  in  every  par- 


ABERRATION.  71 

ticular  with  the  observed  phenomena  of  the  apparent  displace- 
ment of  the  stars,  first  discovered  by  Dr.  Bradley.  These  facts 
show  that  the  aberration  of  light  is  the  true  cause  of  these  pheno- 
mena, and  consequently  establish  at  the  same  time  the  fact  of  the 
progressive  motion  of  light,  and  that  of  the  orbital  motion  of  the 
earth. 

Although  Bradley  derived  from  the  phenomena  of  aberration 
decisive  proof  of  the  progressive  motion  of  light,  it  was  first  dis- 
covered by  Koemer,  a  Danish  astronomer,  in  1675,  from  a  com- 
parison of  observations  upon  the  eclipses  of  Jupiter's  satellites. 

Velocity  of  Light.    "We  have  by  equation  (15), 

vel.  of  earth  :  vel.  of  light ::  sin  20".445  :!::!:  10,088.8 ; 

and  taking  the  velocity  of  the  earth  in  its  orbit  at  65,460  miles 
per  hour,  or  18.1833  miles  per  second,  we  obtain  for  the  velocity 
of  light  183,448  miles  per  second.  The  orbital  velocity  of  the 
earth  here  used  is  that  which  answers  to  the  recent  more 
accurate  determination  of  the  earth's  distance  from  the  sun  (viz. 
91,328,100  miles).  The  result  obtained  for  the  velocity  of  light 
is  nearly  8,000  miles  per  second  less  than  the  former  determina- 
tion, in  which  the  mean  distance  of  the  earth  from  the  sun  was 
taken  a  little  over  95,000,000  miles. 

Light  traverses  the  distance  from  the  sun  to  the  earth  in 
8m.  18s. 


72 


FIGURE  AND  DIMENSIONS  OF  THE  EARTH. 


CHAPTER  V. 

FIGURE  AND  DIMENSIONS  OF  THE  EARTH. — LATITUDE  AND 
LONGITUDE  OF  A  PLACE. 

1O2.  ALTHOUGH  it  is  in  general  sufficient  for  astronomical 
purposes  to  regard  the  earth  as  a  sphere,  still  it  is  necessary  in 
some  cases  of  astronomical  observation  and  computation,  when 
accurate  results  are  desired,  to  take  notice  of  its  deviation  from 
the  spherical  form.  No  account  need,  however,  be  taken  of  the 
irregularities  of  its  surface,  occasioned  by  mountains  and  valleys, 
as  they  are  exceedingly  minute  when  compared  with  the  whole 
extent  of  the  earth.  It  is  to  be  understood,  then,  that  by  the 
figure  of  the  .earth  is  meant  the  general  form  of  its  surface, 
supposing  it  to  be  smooth,  or  that  the  surface  of  the  land  cor- 
responds with  that  of  the  sea. 

1O3.  Method  of  determining  the  Form  of  a  Terre§. 
trial  meridian.  The  figure  of  the  earth  is  ascertained  from  an 
examination  of  the  form  of  the  terrestrial  meridians. 

A  Degree  of  a  terrestrial  meridian  is  an  arc  of  it  corresponding  to 
an  inclination  of  1°  of  the  vertical  lines  at  the  extremities  of  the 
arc.  It  is  also  called  a  Degree  of  Latitude.  Thus,  if  QNE  (Fig. 
39)  represent  a  terrestrial  meridian,  db  will  be  a  degree  of  it  if  it 
be  of  such  length  that  the  angle  aCb  between  the  vertical  lines 
Z'aO,  Z&C,  is  1°. 


The  length  of  a  degree  at  anyplace  will  serve  as  a  mea- 
sure of  the  curvature  of  the  meridian  at  that  place ;  for  it  is 


LENGTH  OF  A  DEGREE.  73 

obvious,  from  considerations  already  presented  (3),  that  the  earth, 
if  not  strictly  spherical,  must  be  nearly  so,  and  therefore  that  a 
degree  ab  (Fig.  39)  may,  with  but  little  if  any  error,  be  considered 
as  an  arc  of  1°  of  a  circle  which  has  its  centre  at  C,  the  point  of 
intersection  of  the  verticals  C#,  C6,  at  the  extremities  of  the  arc. 
The  curvature  will  then  decrease  in  the  same  proportion  as  the 
radius  of  this  circle  increases,  and  therefore  in  the  same  propor- 
tion as  the  length  of  a  degree  increases.  Wherefore,  the  form 
of  a  meridian  may  be  determined  by  measuring  the  length  of  a 
degree  at  various  latitudes. 

1O4.  To  determine  the  Length  of  a  Degree  of  a  Terres- 
trial Meridian.  To  accomplish  this,  we  have, 

(1.)  To  run  a  meridian  line ;  an  operation  which  is  performed 
in  the  following  manner.  An  altitude  and  azimuth  instrument 
(or  some  other  instrument  adapted  to  meridian  observations)  is 
first  placed  at  the  point  of  departure,  and  accurately  adjusted  to 
the  meridian.  A  new  station  is  then  established  by  sighting 
forward  with  the  telescope.  To  this  station  the  instrument  is 
removed,  and  is  there  adjusted  to  the  meridian  by  sighting  back 
to  the  first  station.  A  third  station  is  then  established  by  sight- 
ing forward  with  the  telescope  as  before,  to  which  the  instrument 
is  removed.  By  thus  continually  establishing  new  stations,  and 
carrying  the  instrument  forward,  the  meridian  line  may  be 
marked  out  for  any  required  distance.  The  meridian  adjust- 
ments may  be  corrected  from  time  to  time  by  astronomical  obser- 
vations (42,  58). 

(2.)  To  find  tfie  length  of  the  arc  passed  over.  When  the 
ground  is  level,  the  length  of  the  arc  may  be  directly  measured. 
In  case  the  nature  of  the  ground  is  such  as  not  to  allow  of  a 
direct  measurement,  it  may  be  determined  with  great  precision  by 
means  of  a  base  line  and  a  chain  of  triangles,  the  angles  of  which 
are  measured. 

(3.)  To  find  the  inclination  of  the  verticals  at  the  extreme  stations. 
This  angle  may  be  obtained  by  measuring  the  meridian  zenith 
distances  of  the  same  fixed  star  at  the  two  stations,  correcting 
them  for  refraction,  and  taking  their  difference.  For,  let  0,  (X 
(Fig.  39)  be  the  two  stations  in  question,  Z,  Z'  their  zeniths,  and 
OS,  O'S,  the  directions  of.a  fixed  star,  and  we  shall  have 

pcO'  =  zoi  —  oic  =  zos  —  z'is  =  zqs — z'O's  ; 

that  is,  the  angle  comprised  between  the  verticals  equal  to  the 
difference  of  the  meridian  zenith  distances  of  the  same  star. 

(4.)  The  length  of  an  arc  of  the  meridian,  either  somewhat  greater 
or  less  than  a  degree,  having  been  found  by  tlie  foregoing  operations, 
thence  to  compute  the  kngth  of  a  degree.  Let  N  denote  the  number 
of  degrees  and  parts  of  a  degree  in  the  measured  arc,  A  its  length, 
and  x  the  length  of  a  degree.  Then,  allowing  that  the  earth  for 
an  extent  of  several  degrees  does  not  differ  sensibly  from  a 
sphere,  we  may  state  the  proportion 


FIGURE  AND  DIMENSIONS  OF  THE   EARTH. 


N:  A::l°  :x;  whence  x  =  ±-^_ .... 

105.  Results  of  the  Measure meiits  of  Degrees.  Degrees 
have  been  measured  with  the  greatest  possible  care,  at  various 
latitudes  and  on  various  meridians.  Upon  a  comparison  of  the 
measured  degrees,  it  appears  that  the  length  of  a  degree  increases  as 
we  proceed  from  the  equator  towards  either  pole.  It  follows,  there- 
fore (103),  that  the  curvature  of  a  meridian  is  greatest  at  the 
equator,  and  diminishes  as  the  latitude  increases ;  and  conse- 
quently, that  the  earth  is  flattened  at  the  poles. 

The  fact  of  the  decrease  of  the  curvature  of  a  terrestrial  meri- 
dian from  the  equator  to  the  poles,  leads  to  the  supposition  that 
it  is  an  ellipse,  having  its  major  axis  in  the  plane  of  the  equator 
and  its  minor  axis  coincident  with  the  axis  of  the  earth.  Ana- 
lytical investigations,  founded  on  the  lengths  of  a  degree  in  dif- 
ferent latitudes,  and  on  different  meridians,  have  established  that 
a  meridian  is,  in  fact,  very  nearly  an  ellipse,  and  that  the  earth 
has  very  nearly  the  form  of  an  oblate  spheroid.  The  same  inves- 
tigations have  also  made  known  the  dimensions  of  the  earth. 
The  amount  of  the  oblateness  at  the  poles  is  measured  by  the 
ratio  of  the  difference  of  the  equatorial  and  polar  diameters 
to  the  equatorial  diameter,  which  is  technically  termed  the 
Oblateness  of  the  earth. 


FIG.  40. 


The  form  of  the  earth  has  also  been  determined  by  other 
methods,  which  cannot  here  be  explained.     All  the  results  of 


measurements,  taken  together,  indicate  an  oblateness  of 


1 

299* 
The  following  are  the  dimensions  of  the  earth  in  miles : 

Radius  at  the  equator 3,962.80  miles. 

Eadius  at  the  pole 3,949.55     " 

Difference  of  equatorial  and  polar  radii .      13.25     " 

Radius  at  45°  latitude 3,956.20     " 

Mean  length  of  a  degree  of  meridian . . .      69.048  " 
The  fourth  part  of  a  meridian 6,2 14.33     " 


LATITUDE  OF  A  PLACE.  75 

106.  Inclination  of  Radiu§  to  Vertical  Line.  Owing 
to  the  elliptical  form  of  a  terrestrial  meridian,  the  radius  and  ver- 
tical line  at  a  place  do  not  coincide.  Let  ENQS  (Fig.  40)  repre- 
sent a  terrestrial  meridian.  For  any  point  O  situated  on  this 
meridian,  CO  will  be  the  radius,  and  the  normal  line  ZOB  the  ver- 
tical. The  position  of  the  vertical  line  will  always  be  such  that 
the  apparent  zenith  Z  will  lie  between  the  true  zenith  z  and  the 
elevated  pole  P.  The  inclination  of  the  radius  to  the  vertical  line, 
or  the  angle  COB,  called  the  reduction  of  latitude,  is  greatest  at 
the  latitude  45°,  and  is  there  equal  to  about 


DETERMINATION  OF  THE  LATITUDE  AND  LONGITUDE 
OF  A  PLACE. 

1OT.  The  latitude  and  longitude  of  a  place  ascertain  its  situa- 
tion upon  the  earth's  surface,  and  are  essential  elements  in  many 
astronomical  investigations. 

10V    To  find  the  Latitude  of  a  Place. 

(1.)  By  the  zenith  distances  or  altitudes  of  a  circumpolar  star,  at 
its  upper  and  lower  transits.  The  principle  of  this  method  has 
already  been  stated  (55),  and  represented  to  be  a  particular 
case  of  a  well-known  principle  of  arithmetical  proportions;  the 
following  is  a  detailed  proof  of  it.  Let  Z  (Fig.  41)  repre- 


FIG.  41. 


sent  the  zenith,  HOR  the  horizon,  P  the  pole,  and  S,  S'  the 
points  at  which  the  upper  and  lower  transits  of  a  circumpolar 
star  take  place  ;  HP  will  be  equal  to  the  latitude  (24),  and  ZP 
will  be  equal  to  the  co-latitude.  Now,  we  have 

HP  =  HS  +  PS,  and  HP  =  HS'  —  PS'  =  HS'  —  PS; 


whence,  2HP  =  HS+HS',  or,  HP  =  .  .  .  .(18). 

2 

In  like  manner  we  obtain, 


Wherefore,  let  the  altitudes  of  a  circumpolar  star  at  its  upper  and 
lower  transits  be  measured  and  corrected  for  refraction,  and  their 
half  sum  will  be  the  latitude  ;  or,  let  the  zenith  distances  be 
measured,  and  corrected  for  refraction,  and  their  half  sum  sub- 


76  LATITUDE   AND   LONGITUDE   OF  A  PLACE. 

tracted  from  90°  will  be  the  latitude.  Stars  should  be  selected 
that  have  a  considerable  altitude  at  their  inferior  transit,  for,  the 
greater  is  the  altitude  the  less  is  the  uncertainty  as  to  the  amount 
of  the  refraction.  On  this  principle  the  pole-star  is  to  be  pre- 
ferred  to  all  others. 

(2.)  By  a  single  meridian  altitude  or  zenith  distance.  Let  s,  s',  s" 
(Fig.  10,  p.  21)  be  the  points  of  meridian  passage  of  three  differ- 
ent stars,  the  first  to  the  north  of  the  zenith,  the  second  between 
the  zenith  and  equator,  and  the  third  to  the  south  of  the  equator : 
ZE  =  the  latitude,  and  we  have  for  the  three  stars, 

ZE  =  sE  —  Zs,  ZE  =  s'E-f  Zs',  ZE  =  Zs"  —  s"E. 
Thus,  if  the  zenith  distance  be  called  north  or  south,  according 
as  the  zenith  is  north  or  south  of  the  star  when  on  the  meridian, 
in  case  the  zenith  distance  and  declination  are  of  the  same  name 
I  \  their  sum  will  be  equal  to  the  latitude ;  but  if  they  are  of  differ- 
ent names  their  difference  will  be  the  latitude,  of  the  same  name 
with  the  greater. 

This  method  supposes  the  declination  of  the  body  observed  to 
be  known.  The  declination  of  a  star  or  of  the  sun  at  any  time  is, 
in  practice,  obtained  for  the  solution  of  this  and  other  problems, 
by  the  aid  of  tables,  or  is  taken  by  inspection  from  the  American 
Nautical  Almanac,  or  other  similar  work.  If  the  time  of  the 
meridian  transit  be  known,  the  altitude  may  be  measured  by  a 
sextant  (67).  The  observed  altitude  must  be  corrected  for  refrac- 
tion, and  also  for  parallax  if  the  body  observed  be  the  sun,  or 
moon,  or  either  one  of  the  planets. 

This  method  of  finding  the  latitude  is  the  one  most  generally 
employed  at  sea,  the  sun  being  the  object  observed.  As  the  time 
of  noon  is  not  known  with  accuracy,  several  altitudes  about  the 
time  of  noon  are  taken,  and  the  meridian  altitude  is  deduced  from 
these. 

(3.)  By  the  difference  of  the  meridian  zenith  distances  of  two  stars 
that  cross  the  meridian  near  the  zenith,  on  opposite  sides.  This  is 
Talcotfs  Method  alluded  to  in  connection  with  the  subject  of  the 
zenith  telescope  (59).  It  is  to  be  preferred  to  all  other  methods 
of  determining  the  latitude,  when  the  observer  is  provided  with 
a  zenith  telescope. 

Let  z  be  the  true  zenith  distance  of  the  star  that  passes  to  the 
south  of  the  zenith,  and  §  its  declination ;  z'  and  $  the  true 
zenith  distance  arid  declination  of  the  other  star ;  and  I  the  lati- 
tude of  tbe  station  :  we  then  have 

and  therefore, 

J 

Also,  let  Z  denote  the  apparent  zenith  distance  of  the  star  that 
pass**  to  the  south  of  the  zenith,  r  its  refraction,  and  Z',  /  the 
corresponding  quantities  for  the  other  star ;  then, 


LATITUDE   OF  A   PLACE.  77 


z  =  Z+r,  and  2'  =  Z'  4V; 
and  substituting  in  equation  (a)  we  obtain, 


As  we  may  suppose  the  declinations  of  the  two  stars  to  be 
known,  it  is  then  only  necessary  to  determine  the  values  of  Z  —  Z', 
and  r  —  /.  Now,  if  two  stars  be  selected  whose  zenith  distances 
are  nearly  equal,  their  difference,  Z  —  Z',  can  be  directly  measured 
by  the  micrometer  of  the  zenith  telescope,  and  thus  a  result  ob- 
tained for  the  latitude  free  from  the  instrumental  errors  that 
attend  all  methods  in  which  the  absolute  zenith  distances  are 
measured.  Also,  if  the  selected  stars  pass  the  meridian  near  the 
zenith,  their  refractions  will  be  small,  and  the  amount  of  their 
difference,  r  —  /,  very  minute,  and  liable  to  no  appreciable  uncer- 
tainty. If  77i  and  m'  denote  the  micrometer  readings  in  observ- 
ing the  two  stars,  converted  into  their  equivalent  angular  values, 
equation  (b)  becomes, 

I  =  XS  +  ft+Xm  —  m^+fa  —  r')..  ..(c). 

It  is  here  tacitly  supposed  that  the  micrometer  reading  increases 
with  an  increase  of  zenith  distance.  If  the  reverse  be  true,  the 
second  term  should  be  affected  with  the  negative  sjgn. 

The  only  instrumental  correction  that  is  to  be  applied  to  the 
result  given  by  this  formula,  is  for  any  error  that  may  occur  in 
the  position  of  the  vertical  axis  of  the  zenith  telescope,  when 
either  star  is  observed.  This  is  determined  by  means  of  a  hori- 
zontal level,  attached  to  the  instrument  in  a  position  perpendi- 
cular to  the  horizontal  axis  of  rotation  of  the  telescope  ;  and 
therefore  turning  with  the  instrument  around  the  vertical  axis. 

The  method  of  making  the  observations  is  briefly  as  follows  : 
the  instrument  having  been  previously  adjusted  to  the  meridian, 
the  observer,  by  means  of  the  finding  circle  (p.  32),  sets  the 
telescope  to  the  mean  of  the  zenith  distances  of  the  selected 
pair  of  stars,  and  when  the  preceding  star  has  entered  the  field  fol- 
lows it  with  the  movable  micrometer  wire,  and  bisects  it  as  it 
reaches  the  meridian.  He  then  reads  the  micrometer,  and  also 
the  level  ;  and  turns  the  instrument  around  its  vertical  axis,  180° 
in  azimuth.  When  the  second  star  enters  the  field  of  the  tele- 
scope, it  is  bisected,  like  the  first,  with  the  micrometer-wire  as  it 
reaches  the  meridian.  The  micrometer  and  level  are  then  read 
as  before.  The  micrometer  readings  multiplied  by  the  angular 
value  of  one  revolution  of  the  micrometer-screw,  are  the  values 
of  m  and  m'  in  equation  (c). 

Both  the  north  and  south  ends  of  the  bubble  of  the  level  are 
read  in  each  observation,  and  the  south  end  reading  subtracted 
from  the  north  end  reading.  Half  the  difference  multiplied  by 
the  value  of  one  division  of  the  level  in  seconds  of  arc,  will  be 
the  inclination  of  the  level  to  a  horizontal  line,  in  each  observation. 
The  half  algebraic  sum  of  these  inclinations  for  the  two  observa- 


78  LATITUDE   AND   LONGITUDE  OF  A  PLACE. 

tions,  will  be  the  correction  to  be  applied,  according  to  its  sign, 
to  the  result  obtained  "by  equation  (c),  for  the  deviation  of  the 
vertical  axis  from  the  truly  vertical  position. 

It  is  found  that  the  probable  error,  from  all  causes,  of  a  single 
determination,  by  a  practised  observer,  does  not  exceed  V ;  and 
that  by  continuing  the  observations  upon  a  series  of  pairs  of 
suitably  selected  stars,  for  a  number  of  nights,  the  latitude  of  a 
station  can  be  determined  with  a  probable  error  of  only  0".l, 
which  answers  to  a  distance  on  the  meridian  of  only  ten  feet. 

Reduced  Latitude.  The  astronomical  latitude  being  known, 
the  reduced  latitude  (p.  19,  def.  4)  may  be  obtained  by  subtract- 
ing from  it  the  reduction  of  latitude.  For  if  00  (Fig.  40)  repre- 
sents the  radius,  and  OB  the  vertical,  at  any  place  O,  and  ECQ 
represents  the  terrestrial  equator,  OBQ  will  be  the  astronomical 
latitude,  OCQ  the  reduced  latitude,  and  COB  the  reduction  of 
latitude  ;  and  we  have, 

OBQ  =  OCQ  +  COB,  and  OCQ  =  OBQ  —  COB ....  (20). 

(For  the  practical  method  of  resolving  this  problem,  see  Prob- 
lem XV.) 

1O9.  Longitude  of  a  Place :  — General  Principle. 
There  are  various  methods  of  finding  the  longitude  of  a  place, 
nearly  all  of  which  rest  upon  the  following  principle : 

The  difference  at  any  instant  between  the  local  times  (whether 
sidereal  or  solar),  at  any  place  and  on  the.  first  meridian,  is  ike 
longitude  of  the  place  expressed  in  time ;  and  consequently,  also,  the 
difference  between  the  local  times  at  any  two  places  is  their  difference 
of  longitude  in  time. 

The  truth  of  this  principle  is  easily  established.  In  the  first 
place,  we  remark  that  the  longitude  of  a  place  contains  the  same 
number  of  degrees  and  parts  of  a  degree  as  the  arc  of  the  celes- 
tial equator  comprised  between  the  meridian  of  Greenwich  and 
the  meridian  of  the  place.  Now,  it  is  Oh.  Om.  Os.  of  mean  solar 
time,  or  mean  noon,  at  any  place,  when  the  mean  sun  (36)  is  on 
the  meridian  of  that  particular  place.  Therefore,  as  the  mean 
sun,  moving  in  the  equator,  recedes  from  the  meridian  towards 
the  west  at  the  rate  of  15°  per  mean  solar  hour,  when  it  is  mean 
noon  at  a  place  to  the  west  of  Greenwich,  it  will  be  as  many  hours 
and  parts  of  an  hour  past  mean  noon  at  Greenwich,  as  is  expressed 
by  the  quotient  of  the  division  of  the  arc  of  the  celestial  equator, 
or  its  equal  the  longitude,  by  15.  If  the  place  be  to  the  east 
instead  of  to  the  west  of  Greenwich,  when  it  is  mean  noon  there, 
it  will  be  as  much  before  mean  noon  at  Greenwich  as  is  expressed 
by  the  longitude  of  the  place  converted  into  time  (as  above).  In 
either  situation  of  the  place,  then,  the  principle  just  stated  will 
be  true. 

It  is  plain  that  the  equality  between  the  difference  of  the  times 
and  of  the  longitudes  will  subsist  equally  if  sidereal  instead  of 
solar  time  be  used. 


LONGITUDE  OF  A  PLACE.  79 

110.  To  find  the  Longitude  of  a  Place. 

(1.)  Let  two  observers,  stationed  one  at  Greenwich  and  the  other 
at  the  given  place,  note  die  times  of  the  occurrence  of  some  phenome- 
non which  is  seen  at  the  same  insfant  at  both  places  ;  the  difference 
of  the  observed  times  will  be  the  longitude  in  time.  The  same 
observations  made  at  any  two  places  will  make  known  their 
difference  of  longitude.  If  the  stations  are  not  distant  from  each 
other,  a  signal,  as  the  flashing  of  gunpowder,  or  the  firing  of  a 
rocket,  may  be  observed.  When  they  are  remote  from  each 
other,  celestial  phenomena  must  be  taken.  Eclipses  of  the  satel- 
lites of  Jupiter  and  of  the  moon,  are  phenomena  adapted  to  the 
purpose  in  question.  But  as  in  these  eclipses  the  diminution 
of  the  light  of  the  body  is  not  sudden,  but  gradual,  the  longitude 
cannot  be  obtained  with  very  great  accuracy  from  observations 
made  upon  them. 

(2.)  Transport  a  chronometer  which  has  'been  carefully  adjusted 
to  the  local  time  at  Greenwich,  to  the  place  whose  longitude  is  sought, 
and  compare  the  time  given  by  the  chronometer  with  the  local  time  of 
the  place.  In  the  same  way,  by  transporting  a  chronometer 
from  any  one  place  to  another,  their  difference  of  longitude  may 
be  obtained.  The  error  and  rate  of  the  chronometer  must  be 
determined  at  the  outset,  and  as  often  afterwards  as  circumstances 
will  admit,  that  the  error  at  the  moment  of  the  observation  may 
be  known  as  accurately  as  possible.  To  insure  greater  certainty 
and  precision  in  the  knowledge  of  the  time,  a  number  of  chro- 
nometers are  often  taken,  instead  of  one  only. 

This  method  is  much  used  at  sea ;  the  local  time  being  obtained 
from  an  observation  upon  the  sun  or  some  other  heavenly  body, 
in  a  manner  to  be  hereafter  explained. 

(3.)  Let  the  Greenwich  time  of  the  occurrence  of  some  celestial 
phenomenon  be  computed,  and  note  the  time  of  its  occurrence  at  the 
given  place. 

Eclipses  of  the  sun  and  moon,  and  of  Jupiter's  satellites,  occul- 
tations  of  the  stars  by  the  moon,  and  the  angular  distance  of  the 
moon  from  some  one  of  the  heavenly  bodies,  are  the  phenomena 
employed.  The  Greenwich  times  of  the  beginning  and  end  of 
the  eclipses  of  Jupiters  satellites,  are  published  for  the  solution  of 
the  problem  of  the  longitude  in  the  English  Nautical  Almanac. 
When  the  longitude  is  estimated  from  Washington,  the  Washing- 
ton times  of  the  occurrence  of  the  same  phenomena  may  be  taken 
from  the  American  Nautical  Almanac. 

Eclipses  of  the  sun,  and  occultations  of  the  stars,  furnish  the 
most  exact  determinations  of  the  longitude,  but  they  cannot  be 
used  for  this  purpose  unless  the  longitude  is  already  approxi- 
mately known. 

The  method  of  lunar  distances  is  chiefly  used  at  sea,  and  is 
given  in  detail  in  treatises  on  navigation  and  nautical  astronomy. 

(4.)  Another  and  more  accurate  method  of  determining  the  dif- 


80  LATITUDE   AND   LONGITUDE   OF  A  PLACE. 

ference  of  longitude  of  two  places,  has  recently  been  introduced 
and  perfected  by  American  astronomers.  It  consists  in  the  use 
of  the  electric  telegraph  for  the  transmission  of  signals  from  one 
station  to  the  other,  and  the  introduction  of  the  electro-chrono- 
graph into  the  circuit,  to  measure  off  and  record,  at  each  station, 
the  beats  of  a  sidereal  clock.  The  clock  may  be  at  either  station, 
or  at  some  other  astronomical  station  in  the  circuit.  Its  beats 
are  electrically  transmitted,  and  recorded  upon  a  moving  roll  of 
paper,  adapted  to  the  registers  at  each  station,  in  a  series  of 
equally  distant  dots,  or  in  a  succession  of  equally  distant  breaks 
in  a  continuous  line  (see  Fig.  22,  p.  37).  The  signals  adopted 
are  the  passages  of  a  star  across  the  wires  of  a  transit  instrument. 

The  observer  at  the  most  easterly  station  strikes  his  break- 
circuit  key  as  the  star  passes  each  of  the  wires  in  succession. 
As  the  result,  the  instants  of  these  successive  transits  are  shown 
upon  the  roll  of  paper  at  each  station,  by  breaks  in  the  line  of 
seconds,  falling  between  those  which  indicate  the  seconds.  When 
the  star  reaches  the  meridian  of  the  other  station,  a  similar  set 
of  observations  are  made  by  the  other  observer ;  and  the  instants 
of  the  successive  transits  are  recorded  as  before,  upon  the  roll  of 
paper  at  each  station.  It  then  only  remains  for  each  observer 
to  remove  the  roll  upon  which  the  instants  of  the  passage  of  the 
star  across  the  wires  of  the  transit  instrument  at  each  station  are 
noted,  and  carefully  measure  the  distance  between  each  break 
in  the  time-line,  obtained  by  the  one  set  of  observations,  from 
the  corresponding  break  obtained  by  the  other  set;  then  con- 
vert this  into  the  equivalent  interval  of  time,  and  take  the  mean 
of  all  the  intervals.  This  will  be  his  determination  of  the  dif- 
ference of  longitude  of  the  two  stations,  in  time.  The  mean  of 
the  results  thus  obtained  by  the  two  observers,  is  then  to  be 
taken  as  more  reliable  than  either  of  the  single  determina- 
tions. 

For  greater  accuracy  a  number  of  selected  stars  should  be 
observed.  The  observations  should  also  be  many  times  repeat- 
ed ;  the  clocks  at  the  two  stations  being  alternately  thrown  into 
the  circuit.  The  result  obtained  is  free  from  the  errors  that 
may  exist  in  the  tabular  places  of  the  stars  observed,  and  from 
the  clock  error;  since  neither  of  these  errors  will  affect  the 
intervals  of  time  employed  by  the  stars  in  passing  from  the 
meridian  of  the  one  station  to  that  of  the  other.  But  each 
observer  should  carefully  determine  and  allow  for  the  errors  of 
adjustment  of  his  transit  instrument. 

The  longitudes  of  the  principal  observatories  in  the  United 
States,  and  of  several  important  stations  of  the  United  States 
Coast  Survey,  have  been  very  accurately  determined  by  this 
method. 


OBLIQUITY  OF  THE  ECLIPTIC. 


CHAPTEE  VI. 

APPARENT  MOTION  OF  THE  SUN  IN  THE  HEAVENS. 

111.  The  sun's  declination  and  the  difference  between  the 
right  ascension  of  the  sun  and  that  of  some  fixed  star,  found  from 
day  to  day  (45  and  55)  throughout  a  revolution,  are  the  elements 
from  which  the  circumstances  of  the  sun's  apparent  motion  are 
derived. 

The  curve  on  the  sphere  of  the  heavens,  passing  through  all 
the  successive  positions  thus  determined  from  day  to  day,  is  the 
Ecliptic.  If  we  suppose  it  to  be  a  circle,  as  it  appears  to  be,  its 
position  will  result  from  the  position  of  the  equinoctial  points 
and  its  obliquity  to  the  equator. 

112.  To  find  the  Obliquity  of  the  Ecliptic.     Let  EQA 
(Fig.  42)  represent  the  equator ;  EGA  the  ecliptic ;  and  OC,  OQt 


FIG,  42. 

lines  drawn  through  0,  the  centre  of  the  earth,  and  perpendicu- 
lar to  the  line  of  the  equinoxes,  AOE  :  then  the  angle  COQ  will 
be  the  obliquity  of  the  ecliptic.  This  angle  has  for  its  measure 
the  arc  CQ,  and  therefore  the  obliquity  of  the  ecliptic-is  equal  to  the 
greatest  declination  of  the  sun.  It  can  but  rarely  happen  that  the 
time  of  the  greatest  declination  will  coincide  with  the  instant  of 
noon  at  the  place  where  the  observations  are  made,  but  it  must 
fall  within  at  least  twelve  hours  of  the  noon  for  which  the  ob- 
served declination  is  the  greatest.  In  this  interval  the  change 
of  declination  cannot  exceed  4",  and  therefore  the  greatest 
observed  declination  cannot  differ  more  than  4"  from  the  obli- 
quity. A  formula  has  been  investigated,  which  gives  in  terms 


82  APPARENT  MOTION  OF  THE  SUN. 

of  determinable  quantities  the  difference  between  any  of  the 
greater  declinations  and  the  maximum  declination.  By  reducing, 
by  means  of  this  formula,  a  number  of  the  greater  declinations  to 
the  maximum  declination,  and  taking  the  mean  of  the  indivi- 
dual results,  a  very  accurate  value  of  the  obliquity  may  be 
found. 

The  obliquity  of  the  ecliptic  changes  slightly  from  year  to 
year.  It  is  also  subject  to  a  slight  diminution  from  century 
to  century.  Its  mean  value  at  the  present  date  (Jan.,  1867)  is 
23°  27'  24". 

113.  To  find  the  Position  of  the  Vernal  or  Autumnal 
Equinox. 

(1.)  On  inspecting  the  observed  declinations  of  the  sun,  it  is 
seen  that  about  the  21st  of  March  the  declination  changes  in  the 
interval  of  two  successive  noons  from  south  to  north.  The  ver- 
nal equinox  occurs  at  some  moment  of  this  interval.  Let  RS, 
R'S'  (Fig.  43)  represent  the  declinations  at  the  noons  between 


FIG.  43. 


which  the  equinox  occurs  :  as  one  is  north  and  the  other  south, 
their  sum  (S)  will  be  the  daily  change  of  declination  at  the  time 
of  the  equinox.  Denote  the  time  from  noon  to  noon  by  T. 
Now,  to  find  the  interval  (x)  between  the  noon  preceding  the 
equinox  and  the  instant  of  the  equinox,  state  the  proportion 


S 

on  the  principle  that  the  declination  changes,  for  a  day  or  more, 
proportionally  to  the  time.  Next,  take  the  daily  change  in  right 
ascension  (RR')  on  the  day  of  the  equinox  and  compute  the  value 
of  RE,  by  the  proportion 


x,  or  .  .  ER,  .  RE  . 

o 


add  RE  to  MR,  the  observed  difference  of  right  ascension  (111) 
on  the  day  preceding  the  equinox,  and  the  sum  ME  will  be  the 


POSITION   OF  THE  EQUINOX.  83 

distance  of  the  equinox  from  the  meridian  of  the  star  observed 
in  connection  with  the  sun ;  if  the  star  be  to  the  west  of  the  sun, 
as  in  the  figure. 

The  position  of  the  autumnal  equinox  may  be  found  by  a 
similar  process,  the  only  difference  in  the  circumstances  being 
that  the  declination  changes  from  north  to  south  instead  of  from 
south  to  north. 

If  the  value  of  x  which  results  from  the  first  proportion  be 
added  to  the  time  of  noon  on  the  day  preceding  the  equinox,  the 
result  will  be  the  time  of  the  equinox. 

(2.)  In  the  triangle  RES  (Fig.  42)  we  have  the  angle  RES  =  o 
the  obliquity  of  the  ecliptic,  and  RS  =  D  the  declination  of  the 
sun,  both  of  which  we  may  suppose  to  be  known,  and  we  have 
by  Napier's  first  rule  (Appendix), 

sin  ER  =  tan  (co.  RES)  tan  RS  =  cot  o  tan  D (21), 

whence  we  can  find  ER.  And  by  taking  the  sum  or  difference 
of  ER  and  MR,  according  as  the  star  observed  is  on  the  opposite 
side  of  the  sun  from  the  equinox  or  the  same  side,  we  obtain 
ME  as  before.  If  this  calculation  be  effected  for  a  number  of 
positions,  S,  S',  S",  etc.,  of  the  sun  on  different  days,  and  a  mean 
of  all  the  individual  results  be  taken,  a  more  exact  value  of  ME 
will  be  obtained. 

ME  being  accurately  known,  the  precise  time  of  the  equinox 
may  readily  be  deduced  from  the  observed  daily  variation  of 
right  ascension  on  the  day  of  the  equinox. 

The  calculations  j  ust  mentioned  rest  upon  the  hypothesis  that 
the  ecliptic  is  a  great  circle.  The  close  agreement  which  is 
found  to  subsist  between  the  values  of  ME  deduced  from  obser- 
vations upon  the  sun  in  different  positions,  S,  S',  S",  etc.,  esta- 
blishes the  truth  of  this  hypothesis.  It  is  also  confirmed  by  the 
fact  that  the  right  ascensions  of  the  vernal  and  autumnal  equinox 
differ  by  180°,  since  we  may  infer  from  this  that  the  line  of  the 
equinoxes  passes  through  the  centre  of  the  earth. 

114.  Longitude  of  the  Sun.  The  longitude  of  the  sun 
may  be  expressed  in  terms  of  the  obliquity  of  the  ecliptic  and 
the  right  ascension  or  declination.  In  the  triangle  ERS  (Fig. 
42),  ES  (  =  L)  represents  the  longitude  of  the  sun  supposed  to  be 
at  S,  ER  (  =  R)  its  right  ascension,  and  RS  (  =  D)  its  declina- 
tion. Now,  by  Napier's  first  rule, 

cos  RES=tan  ER  cotES,  orcotES=^?-?— =cosREScotER; 

tan  ER 

thus, 

cot  L  =  cos  Q  cot  R,  or  tan  L  =  ^?_A.    ( .  (22), 

COS    0) 

Also  (Napier's  second  rule,  Appendix), 


84  APPARENT  MOTION  OF  THE  SUN. 

sin  KS  =  cos  (co.  RES)  cos  (co.  ES)  ;  whence,  sin  ES  =  sm 


sin 
or, 

.    -r        sin  D         /00\ 

sm  L  =  —  -----  (23). 

smoj 

With  these  formulae  the  longitude  of  the  sun  may  be  computed 
from  either  its  right  ascension  or  declination.  (See  Prob.  XII., 
Part  III.) 

Formulae  (22)  and  (23)  may  be  written  thus, 

tan  K  =  tan  L  cos  o  ;  sin  D  =  sin  L  sin  o  ----  (24). 

These  formulae  will  make  known  the  right  ascension  and  decli- 
nation of  the  sun,  when  its  longitude  is  given.  (See  Prob.  XL) 
It  will  be  seen  in  the  sequel  that  in  the  present  condition  of 
astronomical  science,  the  longitude  of  the  sun  at  any  assumed 
time  may  be  computed  from  the  ascertained  laws  and  rate  of 
the  sun's  motion. 

115.  Tropical  Year.     The  interval  between  two  successive 
returns  of  the  sun  to  the  same  equinox,  or  to  the  same  longitude, 
is  called  a  Tropical  Year. 

The  interval  between  two  successive  returns  of  the  sun  to  the 
same  position  with  respect  to  the  fixed  stars,  is  called  a  Sidereal 
Year. 

It  appears  from  observation  that  the  length  of  the  tropical 
year  is  subject  to  slight  periodical  variations.  The  period  from 
which  it  deviates  periodically  and  equally  on  both  sides,  is  called 
the  Mean  Tropical  Year.  As  the  changes  in  the  length  of  the 
true  tropical  year  are  very  minute,  the  length  of  the  mean  tropi- 
cal year  is  obviously  very  nearly  equal  to  the  mean  length  of  the 
true  tropical  year,  in  an  interval  during  which  this  passes  one  or 
more  times  through  all  its  different  values.  In  point  of  fact,  it 
may  be  found  with  a  very  close  approximation  to  the  truth  by 
comparing  two  equinoxes  observed  at  an  interval  of  60  or  100 
years. 

According  to  the  most  accurate  determinations,  the  length  of 
the  mean  tropical  year,  expressed  in  mean  solar  time,  is  365d. 
5h.  48m.  46.1s. 

116.  Sun's  Daily  Motion    in  Longitude.      In  a  mean 
tropical  year  the  sun's  mean  motion  in  longitude  is  360°  ;  hence, 
to  find  his  mean  daily  motion  in  longitude  we  have  only  to  state 
the  proportion 

365d.  5h.  48m.  46s.  :  Id.  ::  360  :  x  =  59'  8".33. 
If  from  the  right  ascension  or  declination  of  the  sun,  found 
on  two  successive  days,  the  corresponding  longitudes  be  de- 
duced (equs.  22,  23),  and  their  difference  taken,  the  result 
will  be  the  sun's  daily  motion  in  longitude  at  the  date  of  the 
observations. 


SUN'S  DAILY  MOTION  IX  LONGITUDE.  85 

TJie  sun's  daily  motion  in  longitude  is  not  the  same  throughout 
the  year,  but,  on  the  contrary,  is  continually  varying.  It  gradu- 
ally increases  during  one-half  of  a  revolution,  and  gradually  de- 
creases during  the  other  half,  and  at  the  end  of  the  year  hag 
recovered  its  original  value.  Thus,  the  greatest  and  least  daily 
motions  occur  at  opposite  points  of  the  ecliptic.  They  are,  re- 
spectively, 61'  10"  and  57'  12". 

The  exact  law  of  the  sun's  unequable  motion,  can  only  be  ob- 
tained by  taking  into  account  the  variation  of  his  distance  from 
the  earth  ;  for  the  two  are  essentially  connected  by  the  physical 
law  of  gravitation,  which  determines  the  nature  of  the  earth's 
motion  of  revolution  around  the  sun. 

That  the  distance  of  the  sun  from  the  earth  is  in  fact  subject 
to  a  variation,  may  be  inferred  from  the  observed  fact  that  his 
apparent  diameter  varies.  On  measuring  with  the  micrometer 
the  apparent  diameter  of  the  sun  from  day  to  day  throughout 
the  year,  it  is  found  to  be  the  greatest  when  the  aaily  angular 
motion,  or  in  longitude,  is  the  greatest,  and  the  least  when  the 
daily  motion  is  the  least ;  and  to  vary  gradually  between  these 
two  limits.  Accordingly  the  sun  is  nearest  to  us  when  its  daily 
angular  motion  is  the  most  rapid,  and  farthest  from  us  when  its 
daily  motion  is  the  slowest.  The  greatest  apparent  diameter  of 
the  sun  is  32'  36" ;  and  the  least  apparent  diameter  31'  32". 


86  PKECESSION   OF  THE   EQUINOXES. 


CHAPTER  VII. 

PRECESSION  OF  THE  EQUINOXES. — NUTATION. 

117.  Proof  of  an  Annual  Precession  of  the  Equinoxes. 

The  determination  of  the  position  of  the  vernal  equinox,  consists 
in  deducing  from  the  results  of  certain  observations  the  difference 
between  the  right  ascension  of  the  equinox  and  that  of  one  of  the 
fixed  stars  (113).  This  difference  is  represented  by  ME,  in  Fig. 
42,  and  by  YE  in  Fig.  8.  We  have  seen  (45)  that  when  this 
has  become  known,  the  absolute  right  ascensions  of  all  the  stars 
may  be  determined.  We  have  seen  also  (56),  that  when  the 
right  ascension  and  declination  of  a  star  are  known,  its  longitude 
and  latitude  may  be  computed.  Now,  if  the  position  of  the 
vernal  equinox  be  determined  at  two  epochs  separated  by  a  num- 
ber of  years,  it  is  found  that  the  value  of  ME  has  materially  in- 
creased, if  the  star  s,  observed  with  the  sun,  is  to  the  east  of  the 
equinox  ;  and  decreased  if  the  star  lies  to  the  west  of  the  equinox. 
From  this  fact  we  may  conclude  that  the  equinox  has  a  retro- 
grade motion,  or  towards  the  west,  from  year  to  year. 

Again,  if  the  longitudes  and  latitudes  of  the  same  fixed  stars, 
obtained  as  above,  at  different  periods,  be  compared,  it  is  found 
that  their  latitudes  continue  very  nearly  the  same,  but  that  their 
longitudes  all  increase  at  the  same  mean  rate  of  about  50"  per  year. 
Thus,  EL  (Fig.  42)  represents  the  longitude  of  the  star  s,  and  sL 
its  latitude,  and  it  is  found  that  sL  remains  the  same,  but  that  EL 
increases  at  the  mean  rate  of  50"  per  year.  It  follows,  therefore, 
that  the  vernal  equinox  must  have  an  annual  motion  of  about 
50"  along  the  ecliptic,  in  a  direction  contrary  to  the  order  of  the 
signs,  or  from  east  to  west.  As  it  has  been  ascertained  that  the 
autumnal  is  always  at  the  distance  of  180°  from  the  vernal  equi- 
nox, it  must  have  the  same  motion.  This  retrograde  motion  of 
the  equinoctial  points  is  called  the  Precession  of  the  Equinoxes. 

118.  Ecliptic  Stationary.     As  the  latitude  of  a  star  is  its 
angular  distance  from  the  ecliptic,  it  follows  from  the  circum- 
stance of  the  latitudes  of  all  the  stars  continuing  very  nearly  the 
same,  that  the  ecliptic  remains  fixed,  or  very  nearly  so,  with 
respect  to  the  situations  of  the  fixed  stars. 

The  ecliptic  being  stationary,  it  is  plain  that  the  precession  of 
the  equinoxes  must  result  from  a  continual  slow  motion  of  the 
equator  in  one  direction.  It  appears  from  observation  that  the 
obliquity  of  the  ecliptic,  or  the  inclination  of  the  equator  to  the 


MOTION  OF  THE  POLE  OF  THE  HEAVENS.        87 

ecliptic,  remains  in  the  course   of  this  motion  very  nearly  the 
same. 

119.  Frogre§§ive  Motion  of  the  Pole  of  the  Heavens. 

Since  the  equator  is  in  motion,  its  pole  must  change  its  place  in 
the  heavens.     Let  YLA  (Fig.  44)  represent  the  ecliptic ;  K  its 


stationary  pole ;  P  the  position  of  the  north  pole  of  the  equator, 
or  of  the  heavens,  at  any  given  time,  and  YEA  the  correspond- 
ing position  of  the  line  of  the  equinoxes :  KPL  represents  the 
circle  of  latitude  passing  through  P,  or  the  solstitial  colure. 
Now,  the  point  Y  being  at  the  same  time  in  the  ecliptic  and 
equator,  it  is  90°  distant  from  the  two  points  K  and  P,  the  poles 
of  these  circles ;  therefore,  it  is  the  pole  of  the  circle  KPL 
passing  through  these  points,  and  hence  YL  =  90°.  It  follows 
from  this,  that  when  the  vernal  equinox  has  retrograded  to  any 
point  Y',  the  pole  of  the  equator,  originally  at  P,  will  be  found 
in  the  circle  of  latitude  KP'L'  for  which  Y;L'  equals  90° :  it  will 
also  be  at  the  distance  KP'  from  the  pole  of  the  ecliptic,  equal  to 
KP.  Whence  it  appears  that  the  pole  of  the  equator  has  a  retro- 
grade motion  in  a  small  circle  about  the  pole  of  the  ecliptic,  and 
at  a  distance  from  it  equal  to  the  obliquity  of  the  ecliptic.  As 
the  motion  of  the  equator  which  produces  the  precession  of  the 
equinoxes  is  uniform,  the  motion  of  the  pole  must  be  uniform 
also;  and  as  the  pole  will  accomplish  a  revolution  in  the  same 
time  with  the  equinox,  its  rate  of  motion  must  be  the  same  as 
that  of  the  equinox,  that  is,  50"  of  its  circle  in  a  year.  The 
period  of  revolution  of  the  equinox  and  the  pole  of  the  equator, 
is  about  24,500  years. 

It  is  an  interesting  consequence  of  this  motion  of  the  pole  of 
the  equator  and  heavens,  that  the  pole-star,  so  called,  will  not 
always  be  nearer  to  the  pole  than  any  other  star.  The  pole  is  at 
the  present  time  approaching  it,  and  it  will  continue  to  approach 
it  until  the  present  distance  of  1-J0  becomes  reduced  to  less  than 
i°,  which  will  happen  about  the  year  2100  :*  after  which  it  will 
begin  to  recede  from  it,  and  continue  to  recede,  until  about  the 


88  PRECESSION  OF  THE  EQUINOXES. 

year  3200  another  star  will  come  to  have  the  rank  of  a  pole-star. 
The  motion  of  the  pole  still  continuing,  it  will,  in  the  lapse  of 
centuries,  pass  in  the  vicinity  of  several  pretty  distinct  stars  in 
succession,  and  in  about  12,000  }rears  will  be  within  a  few 
degrees  of  the  star  Yega,  in  the  constellation  of  the  Lyre,  the 
brightest  star  in  the  northern  hemisphere. 

The  present  pole-star  has  held  that  rank  since  the  time  of 
the  celebrated  astronomer  Hipparchus,  who  flourished  about 
120  B.  C.  In  very  ancient  times,  a  pretty  bright  star  in  the 
constellation  of  the  Dragon  (a  Draconis)  was  the  pole-star. 

The  motion  of  the  equator  which  produces  the  precession  of 
the  equinoxes,  must  also  produce  changes  in  the  right  ascensions 
and  declinations  of  the  stars.  These  changes  will  be  different 
according  to  the  situations  of  the  stars  with  respect  to  the  equator 
and  equinoctial  points. 

120.  £ffect  of  Precession  oil  the  Length  of  the  Year. 
The  precession  of  the  equinoxes  makes  the  tropical  year  shorter 
than  the  sidereal  year.     For,  since  the  precession  is  a  retrograde 
movement  of  each  equinox  of  50".24  per  year,  when  the  sun  has 
returned  to  the  same  equinox,  it  will  not  have  accomplished  a 
sidereal  revolution  into  50".24.    The  excess  of  the  sidereal  over 
the  tropical  year  results  from  the  proportion 

69'  S".33::50".24::ld  :  x  =  20m.  23.3s. 

Thus  the  length  of  the  mean  sidereal  year,  expressed  in  mean 
solar  time,  is  365d.  6h.  9m.  9.4s. 

121.  Secular   Diminution    of    the    Obliquity  of   the 
Ecliptic.     The  ecliptic,  although   very  nearly  stationary,  as 
stated  in  Art.  118,  is  not  strictly  so.     By  comparing  the  values 
of  the  obliquity  of  the  ecliptic,  found  at  distant  periods,  it  is 
ascertained  that  it  is  subject  to  a  gradual  diminution  of  46" 
from  century  to  century.     It  appears  from  observation  that  there 
are  minute  secular  changes  in  the  latitudes  of  the  stars,  which 
establish  that  the  progressive  diminution  of  the  obliquity  of  the 
ecliptic  arises  from  a  slow  displacement  of  the  plane  of  the  eclip- 
tic, or  of  the  earth's  orbit,  in  space. 

It  remains  for  us  now  to  take  notice  of  a  minute  inequality  in 
the  motion  of  the  equator  and  its  pole,  which  we  have  thus  far 
overlooked. 

NUTATION. 

122.  Discovery  of  Nutation.     Dr.  Bradley,  in  observing 
the  polar  distance  of  a  certain  star  (/  Draconis)  with  the  view  of 
verifying  his  theory  of  aberration,  discovered  that  the  observed 
polar  distance  did  not  agree  with  the  polar  distance  as  computed 
From  the  results  of  previous  observations,  by  allowing  for  the 
change  due  to  the  precession  in  the  interval ;  the  proper  correc- 
tions for  refraction  and  aberration  having  been  applied  in  both 


ELLIPSE  OF  NUTATION. 


89 


cases.  On  continuing  his  observations  he  found  that  the  polar 
distance  alternately  increased  and  diminished,  and  that  it  returned 
to  the  same  value  in  about  19  years.  These  phenomena  led  him 
to  suppose  that  the  pole,  instead  of  moving  uniformly  in  a  circle 
around  the  pole  of  the  ecliptic,  oscillated  from  the  one  side  to  the 
other  of  a  point  conceived  to  move  in  this  manner. 

123.  Ellipse  of  Nutation.  If  the  pole  has  such  a  motion 
it  is  plain  that,  allowing  the  fact  of  the  earth's  rotation,  it  must 
result  from  a  vibratory  motion  of  the  earth's  axis.  To  this  sup- 
posed vibration  of  the  axis  of  the  earth,  and  consequently  of  that 
of  the  heavens,  Dr.  Bradley  gave  the  name  of  Nutation.  Upon 
a  detailed  examination  of  all  his  observations,  it  appeared  that 
the  oscillation  of  the  pole  did  not  take  place  in  a  right  line,  but 
in  a  minute  ellipse.  The  motion  may  accordingly  be  regarded 
as  a  motion  of  revolution  in  an  ellipse  around  its  centre.  This 
central  point,  about  which  the  pole  revolves,  is  the  mean  position 
of  the  pole,  and  is  called  the  Mean  Pole.  The  direction  of  the 
motion  of  revolution  is  retrograde,  or  from  east  to  west,  and  the 
period  is  about  19  years. 

In  Fig.  45,  pgfg'  represents  the  ellipse  of  nutation,  and  P  the 


mean  pole  ;  the  direction  of  the  motion  of  revolution  being  from 
p  towards/  The  major  axis  g^'  lies  in  the  solstitial  colure  KPL, 
and  is  equal  to  19";  and  the  minor  axis  ff  is  equal  to  14". 
"While  the  true  pole  revolves  in  its  ellipse 
about  the  mean  pole  P,  the  mean  pole  has 
a  uniform  retrograde  movement  in  a  cir- 
cle NPP',  around  the  pole  of  the  ecliptic 
K.  Accordingly  the  pole  has  two  cotem 
poraneous  motions ;  one  in  a  minute  ellipse, 
and  about  its  centre,  and  another  in  a  circle 
of  23J°  radius,  about  the  pole  of  the  eclip- 
tic. Its  actual  motion  must  therefore  be 
in  a  slightly  waving  curve,  passing  alter- 
FIG.  46.  nately  from  one  side  to  the  other  of  this 


90  PRECESSION"  AND  NUTATION". 

circle,  as  shown  in  Fig.  46 ;  in  which,  however,  the  deviations 
from  the  circle  are  greatlj  exaggerated.  The  ellipse  of  nutation 
is  also  greatly  exaggerated  in  Fig.  45. 

124.  Effects  of  Mutation.  As  the  equator  must  move  with  the 
axis  of  the  earth  or  heavens,  nutation  must  change  the  position  of  the  equinox  and  the 
obliquity  of  the  ecliptic.  It  is  plain  that  its  effect  upon  the  position  of  the  equinox 
will  be  to  make  it  oscillate  periodically,  and  by  equal  degrees,  from  one  side  to  the 
other  of  the  position  which  corresponds  to  the  mean  pole ;  and  that  its  effect  upon 
the  obliquity  of  the  ecliptic  will  be  to  make  it  alternately  greater  and  less  than  the 
obliquity  corresponding  to  the  mean  pole.  The  position  of  the  equinox  which  cor- 
responds to  the  mean  pole  is  called  the  Mean  Equinox;  and  the  obliquity  corres- 
ponding to  the  mean  pole  is  called  the  Mean  Obliquity.  Mean  Equator  has  a  like 
signification.  The  real  equinox  and  the  real  equator  are  called,  respectively,  the 
True  Equinox  and  the  True  Equator.  The  actual  obliquity  of  the  ecliptic  is  termed 
the  Apparent  Obliquity. 

In  like  manner,  the  right  ascension,  declination,  etc.,  of  a  star,  referred  to  the 
mean  equator  and  mean  equinox,  are  designated  the  mean  right  ascension,  mean 
declination,  etc. ;  to  distinguish  them  from  the  corresponding  elements  referred  to 
the  true  equator  and  true  equinox.  The  distance  of  the  true  from  the  mean  equi- 
nox in  longitude,  is  called  the  equation  of  the  equinoxes  in  longitude. 


DIFFERENT  KINDS  OF  TIME.  91 


CHAPTER  VIII. 

MEASUREMENT  OF  TIME. 
DIFFEKENT  KINDS  OF  TIME. 

125.  IN  Astronomy,  as  we  have  already  stated,  three  kinds  of 
time  are  used — Sidereal,  True  or  Apparent  Solar,  and  Mean  Solar 
Time  ;  sidereal  time  being  measured  by  the  diurnal  motion  of  the 
vernal  equinox,  true  or  apparent  solar  time  by  that  of  the  sun, 
and  mean  solar  time  by  that  of  an  imaginary  sun  called  the 
Mean  sun,  conceived  to  move  uniformly  in  the  equator  with  the 
real  sun's  mean  motion  in  right  ascension  or  longitude. 

126.  True  Solar  Day.      The   sidereal  day  and  the  mean 
solar  day  are  each  of  uniform  duration,  but  the  length  of  the  true 
solar  day  is  variable,  as  we  will  now  proceed  to  show. 

The  sun's  daily  motion  in  right  ascension,  expressed  in  time, 
is  equal  to  the  excess  of  the  solar  over  the  sidereal  day.  Now 
this  arc,  and  therefore  the  true  solar  day,  varies  from  two 
causes,  viz. : 

(1.)  TJie  inequality  of  the  Sun's  daily  motion  in  longitude. 

(2.)  The  obliquity  of  the  ecliptic  to  the  equator. 

If  the  ecliptic  were  coincident  with  the  equator,  the  daily  arc 
of  right  ascension  would  be  equal  to  the  daily  arc  of  longitude, 
and  therefore  would  vary  between  the  limits  57'  12"  and  61' 10", 
which  would  answer,  respectively,  to  the  apogee  and  perigee. 
But,  owing  to  the  obliquity  of  the  ecliptic,  the  inclination  of  the 
daily  arc  of  longitude  to  the  equator  is  subject  to  a  variation ; 
and  this,  it  is  plain  (see  Fig.  42),  will  be  attended  with  a  varia- 
tion in  the  daily  arc  of  right  ascension.  The  tendency  of  this 
cause  is  obviously  to  make  the  daily  arc  of  right  ascension  least 
at  the  equinoxes,  where  the  obliquity  of  the  arc  of  longitude  is 
greatest,  and  greatest  at  the  solstices,  where  the  obliquity  is 
least. 

127.  Mean  Solar  Time.     As  the  length  of  the  apparent 
solar  day  is  variable,  it  cannot  conveniently  be  employed  for 
the  expression  of  intervals  of  time ;  moreover,  a  clock,  to  keep 
apparent  solar  time,  requires  to  be  frequently  adjusted.     These 
inconveniences  attending  the  use  of  apparent  solar  time,  led 
astronomers  to  devise  a  new  method  of  measuring  time,  to  which 
they  gave  the  name  of  mean  solar  time.     By  conceiving  an 
imaginary  sun  to  move  uniformly  in  the  equator  with  the  real 


92  MEASUREMENT  OF  TIME. 

sun's  mean  motion,  a  day  was  obtained  of  which  the  length  is  inva- 
riable, and  equal  to  the  mean  length  of  all  the  apparent  solar  days 
in  a  tropical  year.  The  point  and  time  of  departure  of  this 
fictitious  sun,  were  also  so  chosen  that  its  distance  from  the  mean 
equinox  would  always  be  equal  to  the  sun's  mean  longitude ; 
the  time  deduced  from  its  position  with  respect  to  the  meridian, 
was  thus  made  to  correspond  very  nearly  with  apparent  solar 
time. 

To  find  the  excess  of  the  mean  solar  day  over  the  sidereal  day, 
we  have  the  proportion 

360°  :  24  sid.  hours  ::  59'  8".33  :  x  =  3m.  56.555s. 

A  mean  solar  day,  comprising  24  mean  solar  hours,  is  there- 
fore 24h.  3m.  56.555s.  of  sidereal  time.  Hence,  a  clock  regulated 
to  sidereal  time  will  gain  3m.  56.555s.  in  a  mean  solar  day. 

To  find  the  expression  for  the  sidereal  day  in  mean  solar 
time,  we  must  uss  the  proportion 

24h.  3m.  56.555s.  :  24h.  ::24h.  :  x  =  23h.  56m.  4.092s. 
The  difference  between  this  and  24  hours  is  3m.  55.908s. ;  and 
therefore,  a  mean  solar  clock  will  lose  with  respect  to  a  sidereal 
clock,  or  with  respect  to  the  fixed  stars,  3m.  55.908s.  in  a  side- 
real day,  and  proportionally  in  other  intervals.  This  is  called 
the  daily  acceleration  of  the  fixed  stars. 

To  express  any  given  period  of  sidereal  time  in  mean  solar  time,  we  must  sub- 
tract for  each  hour  — — : — —=  9.83s.,  and  for  minutes  and  seconds  in  the  same 
proportion.  And,  on  the  other  hand,  to  express  any  given  period  of  mean  solar 
time  in  sidereal  time,  we  must  add  for  each  hour  -^ '—.°JL-=  9.86s.,  and  for  min- 
utes and  seconds  in  the  same  proportion. 

It  is  the  practice  of  astronomers  to  adjust  the  sidereal  clock  to  the  motions  of 
the  true  instead  of  the  mean  equinox.  The  inequality  of  the  diurnal  motion  of 
this  point  is  too  small  to  occasion  any  practical  inconvenience.  Sidereal  time, 
as  determined  by  the  position  of  the  true  equinox,  will  not  deviate  from  the  same 
as  indicated  by  the  position  of  the  mean  equinox,  more  than  2.3s.  hi  19  years. 


CONVERSION  OF  ONE  SPECIES  OF  TIME  INTO  ANOTHER. 

12§.  The  difference  between  the  apparent  and  mean  time  is 
called  the  Equation  of  Time.  The  equation  of  time,  when  known, 
serves  for  the  conversion  of  mean  time  into  apparent,  and  the 
reverse. 

129.  To  find  the  Equation  of  Time.  The  hour  angle 
of  the  sun  (p.  15,  def.  16)  varies  at  the  rate  of  360°  in  a  solar 
day,  or  15°  per  solar  hour.  If,  therefore,  its  value  at  any 
moment  be  divided  by  15,  the  quotient  will  be  the  apparent  time 
at  that  moment.  In  like  manner,  the  hour  angle  of  the  mean 
sun,  divided  by  15,  gives  the  mean  time.  Now,  let  the  circle 


EQUATION  OF  TIME.  93 

YSD  (Fig.  47)  represent  the  equator,  Y  the  vernal  equinox,  M 
the  point  of  the  equator  which  is  on  the  meridian,  and  YS  the 
right  ascension  of  the  sun  ;  and  we  shall  have, 


appar.  time  = 


FIG.  47. 
MS       YM  — YS 


15  15 

Again,  if  we  suppose  S'  to  be  the  position  of  the  mean  sun 
(YS'  being  equal  to  the  mean  longitude  of  the  sun),  we  shall 
have 


mean  time  =         =  ™-VS'; 
15  15 

yg  _ 

thus,  equa.  of  time  =  mean  time  —  ap.  time  =  --  —  .  .  (25)  ; 

15 

or,  the  equation  of  time  is  equal  to  the  difference  between  the  sun's 
true  right  ascension  and  mean  longitude,  converted  into  time. 

This  rule  will  require  some  modification  if  very  great  accuracy  is  desired  ;  for, 
in  seeking  an  expression  for  the  mean  time,  the  circle  YSD  ought  properly  to  be 
considered  as  the  mean  equator,  answering  to  the  mean  pole  (124),  and  the  mean 
longitude  of  the  sun  is  really  estimated  from  the  mean  equinox  V,  and  ought 
therefore  to  be  corrected  by  the  arc  W,  or  the  equation  of  the  equinoxes  hi  right 
ascension. 

The  value  of  the  equation  of  time,  determined  from  formula 
(25),  is  to  be  applied  with  its  sign  to  the  apparent  time  to  obtain 
the  mean,  and  with  the  opposite  sign  to  the  mean  time  to  obtain 
the  apparent. 

A  formula  has  been  investigated,  and  reduced  to  a  table,  which 
makes  known  the  equation  of  time  by  means  of  the  sun's  mean 
longitude.  (See  Table  XII.  ;  also  Art.  158.)  The  value  of  the 
equation  of  time  at  noon,  on  any  day  of  the  year,  is  also  to  be 
found  in  the  tables  of  calculations  for  the  sun,  published  in  the 
Nautical  Almanac.  If  its  value  for  any  other  time  than  noon 
be  desired,  it  may  be  obtained  by  simple  proportion. 

The  equation  of  time  is  zero,  or  mean  and  true  time  are  the 
same  four  times  in  the  year,  viz.  about  the  15th  of  April,  the 
15th  of  June,  the  1st  of  September,  and  the  24th  of  December. 


94  MEASUREMENT  OF  TIME. 

Its.  greatest  additive  value  (to  apparent  time)  is  about  14J 
minutes,  and  occurs  about  the  llth  of  February  ;  and  its  greatest 
subtractive  value  is  about  16|-  minutes,  and  occurs  about  the  3d 
of  November. 

13O.  To  convert  Sidereal  Time  into  Ittean  Time,  and 
Vice  versa. — Making  use  of  Fig.  47,  already  employed,  the  arc  VM,  called  the 
Right  Ascension  of  Mid- Heaven,  expressed  in  time,  is  the  sidereal  time;  VS'  is  the 
right  ascension  of  the  mean  sun,  estimated  from  the  true  equinox,  or  the  mean 
longitude  of  the  sun  corrected  for  the  equation  of  the  equinoxes  in  right  ascension 
(124);  and  MS'  expressed  in  tune,  is  the  mean  time.  Let  the  arcs  VM,  MS',  and 
VS',  converted  into  time,  be  denoted  respectively  by  S,  M,  and  L.  Now, 

VM  =  MS'+VS'; 
or,  S=M  +  L..(26);  andM  =  S  — L..(27). 

If  M  +  L  in  equation  (26)  exceeds  24  hours,  24  hours  must  be  subtracted;  and 
if  L  exceeds  S  in  equation  (27),  24  hours  must  be  added  to  S,  to  render  the  sub- 
traction possible. 

This  problem  may  in  practice  be  solved  most  easily  by  means  of  an  ephemeris  of 
the  sun  (220),  which  gives  the  value  of  S,  or  the  sidereal  time,  at  the  instant  of  mean 
noon  of  each  day,  together  with  a  table  of  the  acceleration  of  sidereal  on  mean 
solar  time,  and  the  corresponding  table  of  the  retardation  of  mean  on  sidereal 
time. 

The  conversion  of  apparent  into  sidereal  time,  or  sidereal  into  apparent  tune, 
may  be  effected  by  first  obtaining  the  mean  time,  and  then  converting  this  into 
sidereal  or  apparent  time,  as  the  case  may  be. 


DETERMINATION  OF  THE  TIME  AND  REGULATION  OF  CLOCKS 
BY  ASTRONOMICAL  OBSERVATIONS. 

131.  The  regulation  of  a  clock  consists  in  finding  its  error 
and  its  rate. 

132.  Mean  Solar  Clock.     The  error  of  a  mean  solar  clock 
is  most  conveniently  determined  from  observations  with  a  transit 
instrument  of  the  time,  as  given  by  the  clock,  of  the  meridian 
passage  of  the  sun's  centre.     The  time  noted  will  be  the  dock- 
time  at  apparent  noon,  and  the  exact  mean  time  at  apparent 
noon  may  be  obtained  by  applying  to  the  apparent  time  (24h., 
or  Oh.  Om.  Os.)  the  equation  of  time  with  its  proper  sign,  which 
may  for  this  purpose  be  taken  from  the  Nautical  Almanac  by 
simple  inspection.     A  comparison  of  the  clock  time  with  the 
exact  mean  time,  will  give  the  error  of  the  clock. 

The  daily  rate  of  a  mean  solar  clock  may  be  ascertained  by 
finding  as  above  the  error  at  two  successive  apparent  noons.  If 
the  two  errors  are  the  same  and  lie  the  same  way,  the  clock  goes 
accurately  to  mean  solar  time  ;  if  they  are  different,  their  differ- 
ence or  sum,  according  as  they  lie  the  same  or  opposite  ways, 
will  be  the  daily  gain  or  loss,  as  the  case  may  be. 

133.  Sidereal  Clock.     The  methods   of  determining   the 
error  and  rate  of  a  sidereal  clock  have  already  been  explained 
(47).     In  practice,  the  apparent  right  ascension  of  the  clock  star 
to  be  observed,  is  taken  from  the  table  of  the  apparent  places  of 
stars,  in  the  Nautical   Almanac,  as  already  intimated.      The 


TIME   BY   OBSERVATIONS   OUT  OF  THE   MERIDIAN. 


95 


method  of  calculating  such  apparent  places  is  given  in  Prob. 


134.   Time  by  OB>§ervation§  out  of  the  Meridian.     In 

default  of  a  transit  instrument,  the  time  may  be  obtained  and 
time-keepers  regulated  by  observations  made  out  of  the  meridian. 
There  are  two  methods  by  which  this  may  be  accomplished, 
called,  respectively,  the  method  of  Single  Altitudes,  and  the 
method  of  Double  Altitudes,  or  of  Equal  Altitudes.  These  we  will 
now  explain. 

.  (1.)  To  determine  the  time  from  a  measured  altitude  of  the  sun,  or 
of  a  star,  its  declination,  and  also  the  latitude  of  the  place  being 
given. 

Let  us  first  suppose  that  the  altitude  of  the  sun  is  taken  ;  cor- 
rect the  measured  altitude  for  refraction  and  parallax,  and  also, 
if  the  sextant  is  the  instrument  used,  for  the  semi-diameter  of 
the  sun.  Then,  if  Z  (Fig.  48) 
represents  the  zenith,  P  the 
elevated  pole,  and  S  the  sun  ; 
in  the  triangle  ZPS  we  shall 
know  ZP=  co-latitude,  PS  = 
co-declination,  and  ZS  =  co-alti- 
tude, from  which  we  may  com- 
pute the  angle  ZPS  (=  P), 
which  is  the  angular  distance 
of  the  sun  from  the  meridian, 
or,  if  expressed  in  time,  the 
time  of  the  observation  from 
apparent  noon;  by  the  following  equations  (A  pp.,  ^Resolution 
of  oblique-angled  spherical  triangles,  Case  1), 

2  k=  ZP+PS+ZS  =  co-lat.+  co-dec.  +  co-alt.  .  .  .(28)  ; 
{     lp  _  sin(£  —  ZP)sin(£  —  PS) 

- 


FiG.  48. 


or 


sin2  JP  = 


sip  (&  —  co-lat.)  sin  (fe  —  co-dec.)  ^ 

sin  (co-lat.)  sin  (co-dec.) 
The  value  of  P  being  derived  from  these  equations  and  con- 
verted into  time  (see  Prob.  III.),  the  result  will  be  the  apparent 
time  at  the  instant  of  the  observation,  if  it  was  made  in  the 
afternoon;  if  not,  what  remains  after  subtracting  it  from  24 
hours  will  be  the  apparent  time.  The  apparent  time  being  found, 
the  mean  time  may  be  deduced  from  it  by  applying  the  equation 
of  time. 

A  more  accurate  result  will  be  obtained  if  several  altitudes  be  measured,  the 
time  of  each  measurement  noted,  and  the  mean  of  all  the  altitudes  taken  and  re- 
garded as  corresponding  to  the  mean  of  the  times.  The  correspondence  will  be 
sufficiently  exact  if  the  measurements  be  all  made  within  the  space  of  10  or  12 
minutes,  and  when  the  sun  is  near  the  prime  vertical.  If  an  even  number  of  altitudes 
be  taken,  and  alternately  of  the  uppe.r  and  lower  limb,  the  mean  of  the  whole  will 
give  the  altitude  of  the  sun's  centre,  without  it  being  necessary  to  know  his  ap- 


96 


MEASUREMENT  OF  TIME. 


parent  semi-diameter.  In  practice,  the  declination  of  the  sun  may  be  taken  for 
the  solution  of  this  problem  from  an  ephemeris  of  the  sun.  For  this  purpose,  the 
time  of  the  observation  arid  the  longitude  of  the  place  must  be  approximately 
known. 

Example.  On  March  20,  1867,  the  following  double  altitudes 
of  the  sun  were  taken  with  a  prismatic  sextant,  at  New  Haven ; 
upper  limb,  64°  12'  0",  64°  21'  35",  64°  33'  0",—  lower  limb,  63° 
38'  50",  63°  51'  0",  63°  58'  5"  ;  the  corresponding  times  of  obser- 
vation, noted  by  a  watch,  were  9h.  6m.  49s.  A.M.,  9h.  7m.  20.5s.,  9h. 
7m.  56s.,  9h.  8m.  29.5s.,  9h.  9m.  7.7s.,  9h.  9m.  31s. ;  the  barometer 
stood  at  30.47in.,  and  the  thermometer  at  34°.  What  was  the 
mean  time  answering  to  the  mean  of  the  times  of  observation  ? 

Mean  of  times  of  observation 9h.  8m.  12.3s.  A.M. 

Long,  of  station  of  observer,  west  of 

Greenwich, 4   51     42 

Corresponding  Greenwich  time 1    59     54.3.   P.M. 

Sun's  dec.  at  that  time,  Am.  Naut.  Aim. . .   0°  11'  37"  S 
Sun's  co-dec.,  or  K  P.  dist 90     11    37 

Mean  of  measured  double  altitudes 64°    5'  45" 

Index  error —  1      3 

2)64      4~42~ 

Appar.  alt.  of  sun's  centre 32°     2'  21" 

Refraction  (Tables  YIIL,  and  IX.)   —  1    37.3 

True  alt.  of  sun's  centre 32      0    43.7 

Lat.  of  station..  41°  18'  37" 

Co-lat 48    41    23 ar.  co.  sin.  0.124276 

Co-dec 90    11    37 ar.  co.  sin.  0.000003 

Co-alt 57    59    16.3 

2)196*  52    16L3 

& .98    26     8~ 

fc  —  co-dec 8    14   31 sin.  9.156408 

£ _ Co-lat 49    44   45. sin.  9.882630 

2)10,108817 

£P  =  22°  26'    7".8 .9.581658 

P  =  44    52   15  .6 
.4 


179m.  29s.  2'" 

2h.  59m.  29.03s. 
12 


64°  11'  45"  

9h.  10m.  15s. 

A.  M. 

64  18  35  

'.  .9   10   36 

64  25  20  .. 

9   10   58 

65  41  40  

9   11   37 

65  49  50  

9  12    4.5 

65  59  50  . 

..9   12   36 

TIME  BY  OBSERVATIONS  OUT  OF  THE  MERIDIAN.  97 

9h.  Om.  30.97s.  A.  M. 
Equa.  of  time .    +  7    41.37 

M.  time  sought  9       8     12.34    A.  M. 
Time  by  watch  9      8     12.3 

Error  of  watch          —  0.04s. 

The  error  of  the  watch,  as  estimated  from  transit  observations, 
was  less  than  Is. 

On  the  same  date,  the  following  measurements  were  made: 

Double  Altitudes  of  Sun.  Times  of  Observation. 

L.  L. 

ILL. 

Barometer,  thermometer,  and  index  error,  same  as  above. 
The  error  of  the  watch,  as  determined  from  these  data,  was 

-f  0.08s. 

In  case  the  altitude  of  a  star  is  taken,  the  value  of  P  derived  from  formula  (30), 
when  converted  into  time,  will  express  the  distance  in  time  of  the  star  from  the 
meridian ;  and  being  added  to  the  right  ascension  of  the  star,  if  the  observation  be 
made  to  the  westward  of  the  meridian,  or  subtracted  from  the  right  ascension 
(increased  by  24h.,  if  necessary)  if  the  observation  be  made  to  the  eastward,  will 
give  the  sidereal  time  of  the  observation. 

(2.)  To  detei'mine  the  time  of  noon  from  equal  altitudes  oftJie  sun> 
the  times  of  the  observations  being  given. 

If  the  sun's  declination  did  not  change  while  he  is  above  the 
horizon,  he  would  have  equal  altitudes  at  equal  times  before  and 
after  apparent  noon.  Hence,  if  to  the  time  of  the  first  observa- 
tion one-half  the  interval  of  time  between  the  two  observation* 
should  be  added,  the  result  would  be  the  time  of  noon,  as  shown 
by  the  clock  or  watch  employed  to  note  the  times  of  the  obser- 
vations. The  deviation  from  12  o'clock  would  be  the  error  of 
the  clock  with  respect  to  apparent  time.  The  difference  between 
this  error  and  the  equation  of  time  would  be  the  error  of  the 
clock  with  respect  to  mean  time. 

But,  as  in  point  of  fact  the  sun's  declination  is  continually 
changing,  equal  altitudes  will  not  have  place  precisely  at  equal 
times  before  and  after  noon,  and  it  is  therefore  necessary,  in  order 
to  obtain  an  exact  result,  to  apply  a  correction  to  the  time  thus 
obtained.  This  correction  is  called  the  Equation  of  Equal  Alti- 
tudes. Tables  have  been  constructed  by  the  aid  of  which  the 
equation  is  easily  obtained.  This  is  at  the  same  time  a  very 
simple  and  quite  accurate  method  of  finding  the  time,  and  the 
error  of  a  clock. 


93  MEASUREMENT  OF  TIME. 

If  equal  altitudes  of  a  star  should  be  observed,  it  is  evident 
that  half  the  interval  of  time  elapsed  would  give  the  time  when 
the  star  passed  the  meridian,  without  any  correction.  From  this 
the  error  of  the  clock  (if  keeping  sidereal  time)  may  be  found,  as 
explained  in  Art.  133. 

THE  CALENDAR. 

135.  Natural  Periods  of  Time.  The  apparent  motions  of 
the  sun,  which  brin^  about  the  regular  succession  of  day  and 
night  and  the  vicissitude  of  the  seasons,  and  the  motion  of  the 
moon  to  and  from  the  sun  in  the  heavens,  attended  with  con- 
spicuous and  regularly  recurring  changes  in  her  disc,  furnish 
three  natural  periods  for  the  measurement  of  the  lapse  of  time  : 
viz.,  1,  the  period  of  the  apparent  revolution  of  the  sun  with 
respect  to  the  meridian,  comprising  the  two  natural  periods  of 
day  and  night,  which  is  called  the  solar  day  ;  2,  the  period  of 
the  apparent  revolution  of  the  sun  with  respect  to  the  equator, 
comprehending  the  four  seasons,  which  is  called  the  tropical 
year  ;  3,  the  period  of  time  in  which  the  moon  passes  through 
all  its  phases  and  returns  to  the  same  position  relative  to  the 
sun,  called  the  lunar  month.  The  day  is  arbitrarily  divided  into 
twenty-four  equal  parts,  called  hours  ;  the  hours  into  sixty  equal 
parts,  called  minutes;  and  the  minutes  into  sixty  equal  parts, 
called  seconds.  The  tropical  year  contains  365d.  5h.  48m.  46s. 
'The  lunar  month  consists  of  about  2HJ  days.  The  week,  con- 
^sisting  of  seven  days,  has  its  origin  in  Divine  appointment  alone. 
A  Calendar  is  a  scheme  for  taking  note  of  the  lapse  of  time,  and 
fixing  the  dates  of  occurrences,  by  means  of  the  four  periods  just 
specified,  viz.,  the  day,  the  week,  the  month,  and  the  year,  or' 
•periods  taken  as  nearly  equal  to  these  as  circumstances  will 
admit.  Different  nations  have,  in  general,  had  calendars  more 
t>r  less  different :  and  the  proper  adjustment  or  regulation  of  the 
calendar  by  astronomical  observation  has  in  all  ages,  and  with 
all  nations,  been  an  object  of  the  highest  importance.  We  pro- 
pose, in  what  follows,  to  explain  only  the  Julian  and  Gregorian 
Calendars. 

138.  The  Julian  Calendar  divides  the  year  into  12  months, 
containing  in  all  365  days.  Now,  it  is  desirable  that  the  calen- 
dar should  always  denote  the  same  parts  of  the  same  season  by 
the  same  days  of  the  same  months :  that,  for  instance,  the  sum- 
mer a<nd  winter  solstices,  if  once  happening  on  the  21st  of  June 
and  21st  of  December,  should  ever  after  be  reckoned  to  happen 
on  the  same  days.;  that  the  date  of  the  sun's  entering  the  equi- 
nox, the  natural  commencement  of  spring,  should,  if  once,  be 
always  on  the  'JOth  of  March.  For  thus  the  labors  of  agriculture, 
•tfhich  really  depend  on  the  situation  of  the  sun  in  the  heavens, 
would  be  simply  and  .truly  regulated  by  the  calendar. 


THE  CALENDAR.  99 

This  would  happen  if  the  civil  year  of  365  days  were  equal  to 
the  astronomical ;  but  the  latter  is  greater ;  therefore,  if  the 
calendar  should  invariably  distribute  the  year  into  365  days, 
it  would  fall  into  this  kind  of  confusion,  that  in  process  of 
time,  and  successively,  the  vernal  equinox  would  happen  on 
every  day  of  the  civil  year.  Let  us  examine  this  more 
nearly. 

Suppose  the  excess  of  the  astronomical  year  above  the  civil  to 
be  exactly  6  hours,  and  on  the  noon  of  March  20th  of  a  certain 
year,  the  sun  to' be  in  the  equinoctial  point ;  then,  after  the  lapse 
of  a  civil  year  of  365  days,  the  sun  would  be  on  the  meridian, 
but  not  in  the  equinoctial  point ;  it  would  be  to  the  west  of  that 
point,  and  would  have  to  move  6  hours  in  order  to  reach  it,  and 
to  complete  the  astronomical  or  tropical  year.  At  the  comple- 
tions of  a  second  and  a  third  civil  year,  the  sun  would  be  still 
more  an'd  more  remote  from  the  equinoctial  point,  and  would  be 
obliged  to  move  for  12,  and  18  hours,  respectively,  before  he 
could  rejoin  it  and  complete  the  astronomical  year. 

At  the  completion  of  a  fourth  civil  year  the  sun  would  be 
more  distant  than  on  the  two  preceding  ones  from  the  equinoc- 
tial point.  In  order  to  rejoin  it,  and  to  complete  the  astronomi- 
cal year,  he  must  move  for  24  hours  ;  that  is,  for  one  whole  day. 
In  other  words,  the  astronomical  year  would  not  be  completed 
till  the  beginning  of  the  next  astronomical  day ;  till,  in  civil 
reckoning,  the  rioon  of  March  list. 

At  the  end  of  four  more  common  civil  years,  the  sun  would 
be  in  the  equinox  on  the  noon  of  March  22d.  At  the  end  of  8 
and  6-4  years,  on  March  23d  and  April  6th,  respectively;  at  the 
end  of  736  years,  the  sun  would  be  in  the  vernal  equinox  on 
September  20th  ;  and  in  a  period  of  1460  years,  the  sun  would 
have  been  in  every  sign  of  the  zodiac  on  the  same  day  of  the 
calendar,  and  in  the  same  sign  on  every  day. 

If  the  excess  of  the  astronomical  above  the  civil  year  were 
really  what  we  have  supposed  it  to  be,  6  hours,  this  confusion  of 
the  calendar  might  be  very  easily  avoided.  It  would  be  neces- 
sary merely  to  make  every  fourth  civil  year  to  consist  of  366 
days ;  and  for  that  purpose  to  interpose,  or  to  intercalate,  a  day 
in  a  month  previous  to  March.  By  \\\\s  intercalation,  what  would 
have  been  March  21st  is  called  March  20th,  and  accordingly 
the  sun  would  be  still  in  the  equinox  on  the  same  day  of  the 
month. 

This  mode  of  correcting  the  calendar  was  adopted  by  Julius 
Caesar.  The  fourth  year  into  which  the  intercalary  day  is  intro- 
duced was  called  Bissextile;  it  is  now  frequently  called  Leap 
year.  The  correction  is  called  the  Julian  correction,  and  the 
length  of  a  mean  Julian  year  is  365d.  6h. 

.By  the  Julian  Calendar,  every  year  that  is  divisible  by  4  is  a  leap 
year,  and  the  rest  common  years. 


100  MEASUREMENT    OF  TIME. 

137*  Reformation  of  the  Calendar.—  Gregorian  Calen- 
dar. The  astronomical  year  being  equal  to  365d.  5h.  48m. 
46.1s,  it  is  less  than  the  mean  Julian  by  llm.  13.9s.,  or  O.OOTSOOd. 
The  Julian  correction,  therefore,  itself  needs  correction.  The 
calendar  regulated  by  it  would,  in  process  of  time,  become  erro- 
neous, and  would  require  reformation. 

The  intercalation  of  the  Julian  correction  being  too  great,  its 
effect  would  be  to  antedate  the  happening  of  the  equinox.  Thus 
(to  return  to  the  old  illustration)  the  sun,  at  the  completion  of 
the  fourth  civil  year,  now  the  Bissextile,  would  have  passed  the 
equinoctial  point  by  a  time  equal  to  four  times  0.007800d. ;  at  the 
end  of  the  next  Bissextile,  by  eight  times  0.007800d. ;  at  the 
end  of  130  years,  by  about  one  day.  In  other  words,  the  sun 
would  have  been  in  the  equinoctial  point  24  hours  previously,  or 
on  the  noon  of  March  19//i. 

In  the  lapse  of  ages  this  error  would  continue  and  be  increased. 
Its  accumulation  in  1300  years  would  amount  to  10  days,  and 
then  the  vernal  equinox  would  be  reckoned  to  happen  on 
March  10th. 

The  error  into  which  the  calendar  had  fallen,  and  would  con- 
tinue to  full,  was  noticed  by  Pope  Gregory  XIII.,  in  1582.  At 
his  time  the  length  of  the  year  was  known  to  greater  precision 
than  at  the  time  of  Julius  Caesar.  It  was  supposed  equal  to 
365d.  5h.  49m.  16.23s.  Gregory,  desirous  that  the  vernal  equi- 
nox should  be  reckoned  on  or  near  March  21st  (on  which  day  it 
happened  in  the  year  325,  when  the  Council  of  Nice  was  held), 
ordered  that  the  day  succeeding  the  4th  of  October,  1582,  instead 
of  being  called  the  5th,  should  be  called  the  15th :  thus  suppress- 
ing 10  days,  which,  in  the  interval  between  the  years  325  and 
1582,  represented  nearly  the  accumulation  of  error  arising  from 
the  excessive  intercalation  of  the  Julian  correction. 

This  act  reformed  the  calendar.  In  order  to  correct  it  in  future 
ages,  it  was  prescribed  that,  at  certain  convenient  periods,  the 
intercalary  day  of  the  Julian  correction  should  be  omitted.  Thus 
the  centurial  years  1700,  1800,  1900,  are,  according  to  the  Julian 
Calendar,  Bissextiles,  but  on  these  it  was  ordered  that  the  inter- 
calary day  should  not  be  inserted  •  inserted  again  in  2000,  but  not 
inserted  in  2100,  2200,  2300;  and  so  on  for  succeeding  centuries. 
By  the  Gregorian  Calendar,  then,  every  centurial  year  that  is  divisible 
by  400  is  a  Bissextile  or  Leap  year,  and  the  others  common  years. 
For  other  than  centurial  years,  the  rule  is  the  same  as  with  the 
Julian  Calendar. 

This  is  a  most  simple  method  of  regulating  the  calendar.  It 
corrects  the  insufficiency  of  the  Julian  correction,  by  omitting  in 
the  space  of  400  years  3  intercalary  days.  It  is  easy  to  estimate 
the  degree  of  its  inaccuracy  ;  for  the  real  error  is  0.007800d.  in  one 
year,  and  400  x  0.007800d.,  or  3.1200d.  in  400  years.  Conse- 
quently 0.120(M.,  or  2h.  52m.  48s.  in  -100  years,  or  1  day  in  3333 


THE  CALENDAR.  101 

years,  is  the  measure  of  the  degree  of  inaccuracy  of  the  Gregorian 
correction. 

The  Gregorian  Calendar  was  adopted  immediately  on  its  pro- 
mulgation, in  all  Catholic  countries,  but  in  those  where  the  Pro- 
testant religion  prevailed  it  did  not  obtain  a  place  till  some  time 
after.  In  England,  "  the  change  of  style,"  as  it  was  called,  took 
place  after  the  2d  of  September,  1752,  eleven  nominal  days  being 
then  struck  out ;  so  that  the  last  day  of  Old  Style  being  the  2d, 
the  first  of  New  Style  (the  next  day)  was  called  the  14th,  instead 
of  the  3d.  The  same  legislative  enactment  which  established  the 
Gregorian  Calendar  in  England,  changed  the  time  of  the  begin- 
ning of  the  year  from  the  25th  of  March  to  the  1st  of  January. 
Thus  the  year  1752,  which  by  the  old  reckoning  would  have 
commenced  with  the  25th  of  March,  was  made  to  begin  with  the 
1st  of  January;  so  that  the  number  of  the  year  is,  for  dates 
falling  between  the  1st  of  January  and  the  25th  of  March,  one 
greater  by  the  new  than  by  the  old  style.  In  consequence  of 
the  intercalary  day  omitted  in  the  year  1800,  there  is  now,  for 
all  dates,  12  days  difference  between  the  old  and  new  style. 

Kussia  is  at  present  the  only  Christian  country  in  which  the 
Gregorian  Calendar  is  not  used. 

The  calendar  months  consist,  each  of  them,  of  30  or  31  davs, 
except  the  second  month,  February,  which,  in  a  common  year, 
contains  28  days,  and  in  a  Bissextile,  29  days;  the  intercalary 
day  being  added  to  the  last  of  this  month. 

To  find  the  number  of  days  comprised  in  any  number  of  civil 
yniirs,  multiply  865  by  the  number  of  years,  and  add  to  the  pro- 
duct as  many  days  as  there  are  Bissextile  years  in  the  period. 


102  MOTIONS  OP  THE  SUN,   MOON,  AND   PLANETS. 


CHAPTER  IX. 

MOTIONS  OF  THE  SUN,  MOON,  AND  PLANETS,  IN  THEIR 

ORBITS. 

KEPLER'S  LAWS. 

138.  The  celebrated  astronomer,  Kepler,  by  examining  the 
observations  upon  the  planets  that  had  been  made  bj  the  re- 
nowned Danish  observer,  Tycho  Brahe,  discovered,  early  in  the 
seventeenth  century,  that  the  motions  of  these  bodies  were  in  con- 
formity with  the  following  laws  : 

(1.)  The  areas  described  by  the  radius-vector  of  a  planet  (or  a  line 
from  the  sun  to  the  planet)  are  proportional  to  the  times. 

(2.)  The  orbit  of  a  planet  is  an  ellipse,  of  which  the  sun  occupies 
one  of  the  foci. 

(8.)  The  squares  of  the  periods  of  revolution  of  the  planets  are 
proportional  to  the  cubes  of  their  mean  distances  from  the  sun,  or  of 
the  semi-major  axes  of  their  orbits. 

These  laws  are  known  by  the  denomination  of  Kepler's  Laws. 
They  were  announced  by  Kepler  as  the  fundamental  laws  of  the 
planetary  motions,  after  a  partial  examination  only  of  these 
motions.  They  have  since  been'completely  verified,  and  shown 
to  hold  good  for  all  the  planets,  including  the  earth.  We  shall 
adopt  the  first  two  laws  for  the  present,  as  hypotheses,  and  show 
in  the  sequel  that  they  are  verified  by  the  results  deducible  from 
them.  These  laws  being  established,  the  third  is  obtained  by 
simply  comparing  the  known  major  axes  and  periods  of  revolution. 

139.  Motion  of  the  Sun  iai  its  Apparent  Orbit.     The 
apparent  motion  of  the  sun  in  space  must  be  subject  to  Kepler's 
first  two  laws ;  for  the  apparent  orbit  of  the  sun  is  of  the  same 
form  and  dimensions  as  the  actual  orbit  of  the  earth,  and  the 
law  and  rate  of  the  sun's  motion  in  its  apparent  orbit  are  the 
same  as  the  law  and  rate  of  the  earth's  motion.     To  establish 
these  two  principles,  let  EE'A  (Fig.  49)  represent  the  ellipse 
orbit  of  the  earth,  and  S  the  position  of  the  sun  in  space.     If  the 
earth  move  from  E  to  E',  as  it  seems  to  remain  stationary  at 
E,  it  is  plain   that  the  sun  will  appear  to  move  from  S  to  S', 
on  the  line  ES'  drawn  parallel  to  E'S  the  actual  direction  of 
the  sun  from  the  earth;  and  at  a  distance  ES'  equal   to  E'S 
the  actual  distance  of  the  sun  from  the  earth.     Thus,  for  every 


KEPLER'S  LAWS.  103 

position  of  the  earth  in  its  orbit,  the  corresponding  apparent  posi- 
tion of  the  sun  is  obtained  by  drawing  a  line  parallel  to  the 
radius  vector  of  the  earth,  and  equal  to  it.  It  follows,  therefore, 
that  the  area  SES'  apparently  described  by  the  radius-vector  of 


FiQ.  49. 

the  sun  (or  a  line  drawn  from  the  sun  to  the  earth)  in  any  inter- 
val of  time,  is  equal  to  the  area  ESE'  actually  described  by  the 
radius-vector  of  the  earth  in  the  same  time;  and  consequently 
that  the  arc  SS'  apparently  described  by  the  sun  in  space,  is  equal 
to  the  arc  EE'  actually  described  in  the  same  time  by  the  earth. 
Whence  we  conclude,  that  the  apparent  motion  of  the  sun  in 
space,  and  the  actual  motion  of  the  earth,  are  the  same  in  every 
particular. 

140.  It  has*been  ascertained  that  the  motion  of  the  moon  in  its 
revolution  around  the  earth,  is  subject  to  the  same  laws  as  the 
motion  of  a  planet  in  its  revolution  around  the  sun.     We  shall 
assume  this  to  be  a  fact,  and  show  that  the  hypothesis  is  verified 
by  the  results  to  which  it  leads. 

141.  Perihelion.— Aphelion.  That  point  of  the  orbit  of  a 
planet,  which  is  nearest  to  the  sun,  is  called  the  Perihelion,  and 
that  point  which  is  most  distant  from  the  sun,  the  Aphelion. 
The  corresponding  points  of  the  moon's  orbit,  or  of  the  sun's 
apparent   orbit,   are   called,   respectively,   the   Perigee  and   the 
Apogee. 

These  points  are  also  called  Apsides  ;  the  former  being  termed 
the  Lower  Apsis,  and  the  latter  the  Higher  Apsis.  The  line  join- 
ing them  is  denominated  the  Line  of  Apsides. 

The  orbits  of  the  sun,  moon,  and  planets,  being  regarded  as 
ellipses,  the  perigee  and  apogee,  or  the  perihelion  and  aphelion, 
are  the  extremities  of  the  major  axis  of  the  orbit. 

142.  Law     of    the    Angular    Motion    of    a    Planet. 
The  law  of  the  angular  motion  of  a  planet  about  the  sun  may  be 
deduced  from  Kepler's  first  law.     Let  PpAp"  (Fig.  50)  repre- 
sent the  orbit  of  a  planet,  considered  as  an  ellipse,  and^>,  p'  two 
positions  of  the  planet  at  two  instants  separated  by  a  short  interr 


104:  MOTIONS  OF  THE   SUN,    MOON,   AND  PLANETS. 

val  of  time  ;  and  let  n  be  the  middle  point  of  the  arc  pp'.  "With 
the  radius  Sn  describe  the  small  circular  arc  Inl',  and  with  the 
radius  Sa,  equal  to  unity,  describe  the  arc  ab.  It  is  plain  that  the 


FIG.  50. 


two  positions  />,  pf  may  be  taken  so  near  to  each  other,  that  the 
area  Spp'  will  be  sensibly  equal  to  the  circular  sector  Sll'.  If  we 
suppose  this  to  be  the  case,  as  the  measure  of  the  sector  is  %lnlf  x 
Sri  =  \ab  X  Sri  (substituting  for  Inl'  its  value,  ab  x  Sra),  we  shall 
have 


area  Spp'  =  \db  x  !Sna. 

When  the  planet  is  at  any  other  part  of  its  orbit,  as  n',  if 
Sp"jp'"  be  an  area  described  in  the  same  interval  of  time  as 
before,  we  shall  have 

area  SpV"  =  fr'V  x  &?'. 
But  these  areas  are  equal  according  to  Kepler's  first  law  :  hence, 

lab  x  S7t2  =  $a'b'  x  Sn/a  ____  (31)  ; 
and  ab  :  a'b'  ::  Sn1*  :  Sn  ; 

that  is,  the  angular  motion  of  a  planet  about  the  sun  for  a  short 
interval  of  time,  is  inversely  proportional  to  the  square  of  the 
radius-vector. 

It  results  from  this  that  the  angular  motion  is  greatest  at  the 
perihelion,  and  least  at  the  aphelion,  and  the  same  at  correspond- 
ing points  on  either  side  of  the  major  axis  :  also,  that  it  decreases 
progressively  from  the  perihelion  to  the  aphelion,  and  increases 
progressively  from  the  aphelion  to  the  perihelion. 

143.  mean  Place.  —  True  Place.  Now  to  compare  the 
true  with  the  mean  angular  motion,  suppose  a  body  to  revolve 
in  a  circle  around  the  sun,  with  the  mean  angular  motion  of  a 
planet,  and  to  set  out  at  the  same  instant  with  it  from  the  peri- 
helion. Let  PMAM'  (Fig.  51)  represent  the  elliptic  orbit  of  the 
planet,  and  PBaB'  the  circle  described  by  the  body.  The  posi- 
tion B  of  this  fictitious  body  at  any  time,  will  be  the  mean  place 
of  the  planet  as  seen  from  the  sun.  The  two  bodies  will  accom- 


MEAN  AND  TRUE  PLACE.  105 

plish  a  semi-revolution  in  the  same  period  of  time,  and  therefore 
be,  respectively,  at  A  and  a  at  the  same  instant ;  for  it  is  obvious 
that  the  fictitious  body  will  accomplish  a  semi-revolution  in  half 
the  period  of  a  whole  revolution,  and  by  Kepler's  law  of  areas,  the 
planet  will  desvibe  a  semi-ellipse  in  half  the  time  of  a  revolution. 
At  the  outset,  the  motion  of  the 
planet  is  the  most  rapid  (142), 
but  it  continually  decreases  until 
the  planet  reaches  the  aphelion, 
while  the  motion  of  the  body 
remains  constantly  equal  to  the 
mean  motion.  The  planet  will 
therefore  take  the  lead,  and  its 
angular  distance  ^>SB  from  the 
body  will  increase  until  its  mo- 
tion becomes  reduced  to  an 
equality  with  the  mean  motion ;  FIG.  51. 

after  which  it  will  decrease  until 

the  planet  has  reached  the  aphelion  A,  where  it  will  be  zero.  In 
the  motion  from  the  aphelion  to  the  perihelion,  the  angular  velo- 
city of  the  planet  will  at  first  be  less  than  that  of  the  body  (142), 
but  it  will  continually  increase,  while  that  of  the  body  will  re- 
main unaltered:  thus,  the  body  will  now  get  in  advance  of  the 
planet,  and  their  angular  distance  jt/SB'  will  increase,  as  before, 
until  the  motion  of  the  planet  again  attains  to  an  equality  with, 
the  mean  motion,  after  which  it  will  decrease  as  before,  until  it 
again  becomes  zero  at  the  perihelion. 

It  appears,  then,  that  from  the  perihelion  to  the  aphelion  the 
true  place  is  in  advance  of  the  mean  place  /  and  that  from  the 
aphelion  to  the  perihelion,  on  the  contrary,  the  mean  place  is  in 
advance  of  the  true  place. 

The  angular  distance  of  the  true  place  of  a  planet  from  its 
mean  place,  as  it  would  be  observed  from  the  sun,  is  called  the 
Equation  of  the  Centre.  Thus,  jySB  is  the  equation  of  the  cen- 
tre corresponding  to  the  particular  position  p  of  the  planet.  It  is 
evident,  from  the  foregoing  remarks,  that  the  equation  of  the 
centre  is  zero  at  the  perihelion  and  aphelion,  and  greatest  at  the 
two  points,  as  M  and  M',  where  the  planet  has  its  mean  motion. 
The  greatest  value  of  the  equation  of  the  centre  is  called  the 
Greatest  Equation  of  the  Centre. 

As  the  laws  of  the  motion  of  the  moon  (140),  and  of  the  appa- 
rent motion  of  the  sun  (139),  are  the  same  as  those  of  a  planet, 
the  principles  established  in  the  two  preceding  articles  are  as 
applicable  to  these  bodies  in  their  revolution  around  the  earth, 
as  to  a  planet  in  its  revolution  around  the  sun. 


106  MOTIONS   OF  THE  SUN,   MOON,   AND   PLANETS. 


DEFINITIONS  OF  TERMS. 

144.  (1.)  The  Geocentric  Place  of  a  body  is  its  place  as  seen 
from  the  earth. 

(2.)  The  Heliocentric  Place  of  a  body  is  its  place  as  it  would 
be  seen  from  the  sun. 

(3.)  Geocentric  Longitude  and  Latitude  appertain  to  the  geo- 
centric pLice,  and  Heliocentric  Longitude  and  Latitude  to  the 
heliocentric  place. 

(4.)  Two  heaventy  bodies  are  said  to  be  in  Conjunction  when 
their  longitudes  are  the  same,  and  to  be  in  Opposition  when  their 
longitudes  differ  by  180°.  When  any  one  heavenly  body  is  in 
conjunction  with  the  sun,  it  is,  for  the  sake  of  brevity,  said  to  be 
in  Conjunction  /  and  when  it  is  in  opposition  to  the  sun,  to  be 
in  Opposition. 

The  planets  Mercury  and  Yenus,  allowing  that  their  distances 
from  the  sun  are  each  less  than  the  earth's  distance  (18),  can 
never  be  in  opposition.  But  they  may  be  in  conjunction,  either 
by  being  between  the  sun  and  earth,  or  by  being  on  the  op- 
posite side  of  the  sun.  In  the  former  situation  they  are  said 
to  be  in  Inferior  Conjunction,  and  in  the  latter  in  Superior 
Conjunction. 

(5.)  'A  Synodic  Revolution  of  a  body  is  the  interval  between 
two  consecutive  conjunctions  or  oppositions. 

For  the  planets  Mercury  and  Venus  a  synodic  revolution  is 
the  interval  between  two  consecutive  inferior  or  superior  con- 
junctions. 

(6.)  The  Periodic  Time  of  a  planet  is  the  period  of  time  in 
which  it  accomplishes  a  revolution  around  the  sun. 

(7.)  The  Nodes  of  a  planet's  orbit,  or  of  the  moon's  orbit,  are 
the  points  in  which  the  orbit  cuts  the  plane  of  the  ecliptic.  The 
node  at.  which  the  planet  passes  from  the  south  to  the  north  side 
of  the  ecliptic  is  called  th.e  Ascending  Node,  and  is  designated 
by  the  character  Q.  The  other  is  called  the  Descending  Node, 
and  is  marked  y. 

(S)  The  Eccentricity  of  an  elliptic  orbit  is  the  ratio  which  the 
distance  between  the  centre  of  the  orbit  and  either  focus  bears  to 
the  semi-major  axis. 

145.  To  illustrate  these  Definitions,  let  EE'E"  (Fig   52) 
represent  the  orbit  of  the  earth ;  CT)C  the  orbit  of  Venus,  or 
Mercury,  which  we  will  suppose,  for  the  snke  of  sirnplidtv,  to 
lie  in  the  plane  of  the  ecliptic  or  of  the  earth's  orbit;  LNP  a 
part  of  the  orbit  of  Mars,  or  of  any  other  planet  more  distant 
from  the  sun  S  than  the  earth  is  ;  and  ANB  a  part  of  the  projec- 
tion of  this  orbit  on  the  plane  of  the  ecliptic.     N  or  &  will  re- 
present the  ascending  node  of  the  orbit ;  and  the  descending 
node  will  be  diametrically  opposite  to  this  in  the  direction  Sn'. 


DEFINITIONS  OF  TEEMS. 


107 


Also  let  SY  be  the  direction  of  the  vernal  equinox,  as  seen  from 
the  sun,  and  EV,  E'Y  the  parallel  directions  of  the  same  point, 
as  seen  from  the  earth  in  the  two  positions  E  and  E' ;  and  P 
being  supposed  to  be  one  position  of  Mars  in  his  orbit,  let  p  be 


the  projection  of  that  position  on  the  plane  of  the  ecliptic.  The 
heliocentric  longitude  and  latitude  of  Mars,  in  the  position  P, 
are  respectively  YS/?  and  PS^?;  and  if  the  earth  be  at  E,  his 
geocentric  longitude  and  latitude  are  respectively  YE/?  and  PE/>. 
If  we  suppose  that  when  Mars  is  at  P  the  earth  is  at  E',  he  will 
be  in  conjunction  •  and  if  we  suppose  the  earth  to  be  at  E"'  he 
will  be  in  opposition.  Again,  if  we  suppose  the  earth  to  be  at 
E,  and  Yen  us  at  C,  she  will  be  in  superior  conjunction  ;  but 
if  we  suppose  that  Yen  us  is  at  C'  at  the  time  that  the  earth  is  at 
E,  she  will  be  in  inferior  conjunction.  The  term  inferior  is 
used  here  in  the  sense  of  lower  in  place,  or  nearer  the  earth  ; 
and  superior  \\\  the  sense  of  higher  in  place,  or  farther  from  the 
earth.  Since  the  earth  and  planets  are  continually  in  motion,  it 
is  manifest  that  the  positions  of  conjunction  and  opposition  will 
recur  at  different  parts  of  the  orbit,  and  in  process  of  time  in 
every  variety  of  position.  The  time  employed  by  a  planet  in 
passing  around  from  one  position  of  conjunction  or  opposition  to 
another,  called  the  synodic  revolution,  is,  for  the  same  reason, 
longer  than  tine  periodic  time,  or  time  of  passing  around  from 
one  point  of  the  orbit  to  the  same  again. 


108  MOTIONS  OF  THE   SUN,    MOON,   AND  PLANETS. 


ELEMENTS  OF  THE  ORBIT  OF  A  PLANET. 

146.  To  have  a  complete  knowledge  of  the  motions  of  the 
planets,  so  as  to  be  able  to  calculate  the  place  of  any  one  of  them 
at  any  assumed  time,  it  is  necessary  to  know  for  each  planet,  in 
addition  to  the  laws  of  its  motion  discovered  by  Kepler,  the 
position  and  dimensions  of  its  orbit,  its  mean  motion,  and  its 
place  at  a  specified  epoch.     These  necessary  particulars  of  infor- 
mation are  subdivided  into  seven  distinct  elements,  called  the 
Elements  of  the  Orbit  of  a  Planet,  which  are  as  follows : 

(1.)  The  longitude  of  the  ascending  node. 

(2.)  The  inclination  of  the  plane  of  the  orbit  to  the  plane  of 
the  ecliptic,  called  the  inclination  of  the  orbit. 

(3.)  The  mean  distance  of  the  planet  from  the  sun,  or  the  semi- 
major  axis  of  its  orbit. 

(4.)  The  eccentricity  of  the  orbit. 

(5.)  The  heliocentric  longitude  of  the  perihelion. 

(6.)  The  epoch  of  the  perihelion  passage  of  the  planet,  or 
instead,  the  mean  longitude  of  the  planet  at  a  given  epoch. 

(7.)  The  periodic  time  of  the  planet. 

The  first  two  ascertain  imposition  of  the  plane  of  the  planet's 
orbit ;  the  third  and  fourth,  the  dimensions  of  the  orbit ;  the 
fifth,  the  position  of  the  orbit  in  its  plane  ;  the  sixth,  the  place 
of  the  planet  at  a  given  epoch ;  and  the  seventh,  its  mean  rate 
of  motion. 

The  elements  of  the  earth's  orbit,  or  of  the  surfs  apparent 
orbit,  are  but  five  in  number;  the  first  two  of  the  above-men- 
tioned elements  being  wanting,  as  the  plane  of  the  orbit  is  coin- 
cident with  the  plane  of  the  ecliptic. 

The  elements  of  the  modi's  orbit  are  the  same  with  those  of 
a  planet's  orbit,  it  being  understood  that  the  perigee  of  the  moon's 
orbit  answers  to  the  perihelion  of  a  planet's  orbit,  and  that  the 
geocentric  longitude  of  the  perigee  and  the  geocentric  longitude 
of  the  node  of  the  moon's  orbit  answer,  respectively,  to  the  helio- 
centric longitude  of  the  perihelion  and  the  heliocentric  longitude 
of  the  node  of  a  planet's  orbit. 

147.  The  Linear  Unit  adopted,  in  terms  of  which  the  semi- 
major  axes  and  radius- vectors  of  the  planetary  orbits  are  ex- 
pressed, is  the  mean  distance  of  the  sun  from  the  earth,  or  the 
semi»major  axis  of  the   earth's  orbit.     When  thus   expressed, 
these  lines  are  readily  obtained  in  known  measures  whenever 
the  mean  distance  of  the  sun  becomes  known.     The  lines  of 
the  moon's  orbit  are  found  in  terms  of  the  moon's  mean  distance 
from  the  earth,  as  unity. 


MEAN  DISTANCE  OF  THE  SUN.  109 


DETERMINATION  OF  THE  ELEMENTS   OF  THE   SUN'S  APPA- 
RENT ORBIT,  OR  OF  THE  EARTH'S  REAL  ORBIT. 

MEAN  MOTION. 

14§.  The  sun's  mean  daily  motion  in  longitude  results  from 
the  length  of  the  mean  tropical  year  obtained  from  observa- 
tion (115). 

SEMI-MAJOR  AXIS. 

149.  As  we  have  just  stated,  the  semi-major  axis  of  the  sun's 
apparent  orbit,  is  the  linear  unit  in  terms  of  which  the  dimensions 
of  the  planetary  orbits  are  expressed.  Its  absolute  length  is 
computed  from  the  mean  horizontal  parallax  of  the  sun. 

The  Horizontal  Parallax  of  a  body  being  given,  to 
find  its  Distance  from  tlie  Earth.  AYe  have  (equation  7, 
Art.  88) 


sin  H 

where  H  represents  the  horizontal  parallax  of  the  body,  D  its 
distance  from  the  centre  of  the  earth,  and  R  the  radius  of  the 
earth.  The  parallax  of  all  the  heavenly  bodies,  with  the  excep- 
tion of  the  moon,  is  so  small,  that  it  may,  without  material  error, 
be  taken  in  this  equation  in  place  of  its  sine.  Thus, 


sm  H  H 

Again,  since  6.2831853  is  the  length  of  the  circumference  of  a 
circle  of  which  the  radius  is  1,  and  1298000  is  the  number  of 
seconds  in  the  circumference,  we  have  6.2831853  :  1  ::  1296000": 
x  =  206264."806  =  the  length  of  the  radius  (1)  expressed  in 
seconds.  Hence,  if  the  value  of  H  be  expressed  in  seconds, 

-P.      p  206264."806         ,w 
D  =  K  -  —  -----  (66). 

150.  Determination  of  the  Sim's  Mean  Horizontal 
Parallax.  In  the  determination  of  the  sun's  parallax,  by  the 
process  of  Art.  90.  on  error  of  2"  or  3",  equal  to  about  one- 
fourth  of  the  whole  parallax,  may  be  committed,  so  that  the  dis- 
tance of  the  sun,  as  deduced  by  equation  (33)  from  his  parallax 
found  in  that  manner,  may  be  in  error  by  an  amount  equal  to 
one-fourth  or  more  of  the  true  distance.  There  are  more  accu- 
rate methods  of  obtaining  the  sun's  parallax.  By  one  method, 
which  will  be  noticed  in  another  connection,  the  equatorial  paral- 
lax of  the  sun  (92)  was  deduced  from  certain  observations  made 
upon  Venus,  when  seen  to  pass  between  the  sun  and  earth,  in 
176  L  and  1769,  and  the  value  8".58  obtained.  This  is  the  value 
of  the  sun's  equatorial  horizontal  parallax  which  has  been  uni- 


110  APPARENT   MOTION   OF  THE  SUN, 

versally  adopted  until  within  a  very  few  years.  Quite  recently, 
several  different  determinations  have  been  made  of  this  impor- 
tant element,  by  independent  astronomical  methods.  The  differ- 
ent values  obtained  fall  between  8  ".93  and  8  ".97,  the  mean  of 
which  is  8".95.  One  of  these  has  been  the  deduction  of  the 
solar  parallax,  by  the  process  of  Art.  90,  from  the  parallax  of 
Mars  determined  by  direct  observations  at  the  opposition  of 
this  planet,  in  1862,  when  its  distance  from  the  earth  attained 
its  minimum  value.  This  deduction  was  easily  effected,  since,  as 
will  appear  in  the  next  Chapter,  the  theory  of  the  orbital  motions 
of  the  planets  would  give  the  distance  of  Mars  from  the  earth 
at  the  epoch  of  the  observations,  in  terms  of  the  mean  distance 
of  the  sun  from  the  earth  as  the  linear  unit  (147).  The  mean 
of  two  results  obtained  from  the  observations  made  by  two  sets 
of  observers,  at  localities  remote  from  each  other,  is  8".9o. 
This  value,  which  is  the  mean  of  all  the  results,  has  been  defini- 
tively fixed  upon  in  the  most  approved  Solar  Tables  (Leverrier's) ; 
and  has  since  been  adopted  in  the  English  Nautical  Almanac  for 
1870.  It  may  be  relied  upon  as  exact  to  within  a  small  fraction 
of  a  second. 

151.  Calculation  of  Sim's  Mean  Distance.  We  have, 
then,  for  the  sun's  mean  distance  from  the  earth,  or  the  semi- 
major  axis  of  its  orbit, 

D  =  ft  206264^.806  =  23046.347  B  =  91,328,064  miles; 

H 
taking  for  B  the  equatorial  radius  of  the  earth,  3962.80  miles. 


ECCENTRICITY. 

152.  First  Method.  By  the  greatest  and  least  daily  motions 
in  longitude.  We  have  already  explained  (116)  the  mode  of 
deriving  from  observation  the  sun's  motion  in  longitude  from 
day  to  day.  Now,  let  v  =  the  greatest  daily  motion  in  longitude  ; 
v'=.  the  least  daily  motion  in  longitude  ;  r  =  the  least  or  peri- 
gean  distance  of  the  sun  ;  and  r'  the  greatest  or  apogean  dis- 
tance ;  and  we  shall,  have,  by  the  principle  of  Art.  142, 


whence,    r'  +  r  :  r'  —  r  ::  -\/  v  -f  V~v'  :  y~v~  —  V~v't 


\       but, 


:  ,'  -r  :  :  :  VT-  V  7  : 

*  2 


_  =  semi-major  axis  =  1 ;  and  r' —  r  =  2(eccentricity)  =  2  e ; 


ECCENTRICITY  OF  SUN'S  APPARENT  ORBIT.  Ill 

V  v' 


thus,  1  :  2«  :  : 

and 


-- 

The  greatest  and  least  daily  rnotioifs  are,  respectively  (at  a 
ean),  6T.167  and  57'.200.     Substituting,  we  have 


V  v  -{-  ^  v' 

e  greatest  and  le 
mean 

e  =  0.016761 

The  eccentricity  may  also  be  obtained  from  the  greatest  and 
least  apparent  diameters,  by  a  process  similar  to  the  foregoing, 
on  the  principle  that  the  distances  of  the  sun  at  different  times 
are  inversely  proportional  to  its  corresponding  apparent  dia- 
meters (116). 

153.  Second  Mtethod.  By  the  greatest  equation  of  the  centre. 
(1.)  To  find  the  greatest  equation  of  Hie  centre.  Let  L=the  true  longitude,  and 
M  =  the  mean  longitude,  at  the  time  the  true  and  mean  motions  are  equal  between 
the  perigee  and  apogee  (143)  ;  L'  =  the  true  longitude  and  M'  =  the  mean  longi- 
tude, when  the  motions  are  equal  between  the  apogee  and  perigee  ;  and  E  =  the 
greatest  equation  of  the  centre.  Then  (  !  43) 

L  =  M  +  E,  and  L'  =  M'  —  E  ; 
whence,  L'  —  L  =  M'  —  M  —  2E, 

and 


About  the  time  of  the  greatest  equation  the  sun's  true  motion,  and  consequently 
the  equation  of  the  centre,  continues  very  nearly  the  same  for  two  or  three  days  ; 
we  may  therefore,  with  but  slight  error,  take  the  noon,  when  the  sun  is  on  either 
side  of  the  line  of  apsides,  that  separates  the  two  days  on  which  the  motions  in 
longitude  are  most  nearly  equal  to  59'  8",  as  the  epoch  of  the  greatest  equation. 

The  longitude  L  or  L'  at  either  epoch  thus  ascertained,  results  from  the  observed 
right  ascension  and  declination.  M'  —  M  =  the  mean  motion  in  longitude  in  the 
interval  of  the  epochs,  and  is  found  by  multiplying  the  number  of  mean  solar  days 
and  fractions  of  a  day  comprised  in  the  interval  by  59'  8".330,  the  mean  daily 
motion  in  longitude. 

For  example  :  from  observations  upon  the  sun.  made  by  Dr.  Maskelyne,  in  the 
year  1775,  it  is  ascertained  in  the  manner  just  explained,  that  the  sun  was  near  its 
greatest  equation  at  noon,  or  at  Oh.  3m.  35s.  mean  solar  time,  on  the  2d  April,  and 
at  noon  on  the  31st,  or  at  23h.  49m.  35s.  mean  solar  time,  on  the  30th  of  Septem- 
ber. The  observed  longitudes  were,  at  the  first  period  12°  33'  39".06,  and  at  the 
second  188°  5'  44".45.  The  interval  of  time  oetween  the  two  epochs  is  182d,  — 
14m. 

Mean  motion  in  182d.  —14m  ....................  179°  22'  41".56 

Difference  of  two  longitudes  .....................  175    32     5  .39 


Difference 2)    3    5036.17 


Greatest  equation  of  centre 1    55   ]  8  .08 

More  accurate  results  are  obtained  by  reducing  observations  made  during  seve- 
ral days  before  and  after  the  epoch  of  the  greatest  equation,  and  taking  the  mean 
of  the  different  values  of  the  greatest  equation  thus  obtained.  According  to  M. 
Delambre,  the  greatest  equation  was  in  1775,  1°  55'  31".66. 

(2.)  The  eccentricity  of  an  orbit  may  be  derived  from  the  greatest  equation  of 
the  centre  by  means  of  the  following  formula  : 


112  APPARENT  MOTION  OF  THE  SUN. 


2          3.2"         3.5.2 


•fjl 

in  which  K  stands  for  the  expression  ----  (E  being  the  greatest  equation 

of  the  centre).  In  the  case  of  the  sun's  orbit,  K  being  a  small  fraction,  all  its 
powers  beyond  the  first  may  be  omitted.  Thus,  retaining  only  the  first  term  of  the 
series,  and  taking  E  =  1*  55'  31".66  the  greatest  equation  in  1775,  we  have 

'"          =.016803. 


2        2  x  570.2957795 
154.    Equation  of  Centre    depends    on    Eccentricity. 

It  appears  from  the  law  of  the  angular  velocity  of  a  revolving 
body,  investigated  in  Art.  142,  that  the  amount  of  the  propor- 
tional variation  of  this  velocity,  which  obtains  in  the  course  of 
a  revolution,  depends  altogether  upon  the  amount  of  the  propor- 
tional variation  of  distance,  or,  in  other  words,  upon  the  eccen- 
tricity of  the  orbit  (def.  8,  p.  106).  It  follows,  therefore,  that  the 
amount  of  the  greatest  deviation  of  the  true  place  from  the  mean 
place,  that  is,  of  the  greatest  equation  of  the  centre  (143),  must 
depend  upon  the  value  of  the  eccentricity.  If  the  eccentricity  be 
great,  the  greatest  equation  of  the  centre  will  have  a  large  value  ; 
and  if  the  eccentricity  be  equal  to  zero,  that  is,  if  the  orbit  be  a 
circle,  the  equation  of  the  centre  will  also  be  equal  to  zero,  or 
the  true  and  mean  place  will  continually  coincide. 

If  either  of  the  two  quantities,  the  greatest  equation  and  the 
eccentricity,  be  known,  the  other  will  then  become  determinate; 
and  formula  have  been  investigated  which  make  known  either 
one  when  the  other  is  given.  Equation  36  is  the  formula  for 
the  eccentricity. 

From  observations  made  at  distant  periods  it  is  discovered 
that  the  equation  of  the  centre,  and  consequently  the  eccentricity, 
is  subject  to  a  continual  slow  diminution.  The  amount  of  the 
diminution  of  the  greatest  equation,  in  a  century,  is  17".6. 


LONGITUDE  AND  EPOCH  OF  THE  PERIGEE. 

155.  Methods  of  Determination.  As  the  sun's  angular 
velocity  is  the  greatest  at  the  perigee,  the  longitude  of  the  sun 
at  the  time  its  angular  velocity  is  greatest  will  be  the  longitude 
of  the  perigee.  The  time  of  the  greatest  angular  velocity  may 
be  easily  obtained,  within  a  few  hours,  by  means  of  the  daily 
motions  in  longitude  derived  from  observation  (116). 

The  wore  accurate  method  of  determining  the  longitude  and 
epoch  of  the  perigee,  rests  upon  the  principle  that  the  apogee  and 
perigee  are  the  only  two  points  of  the  orbit  whose  longitudes 
differ  by  180°,  in  passing  from  one  to  the  other  of  which  the  sun 
employs  half  a  year.  This  principle  may  be  inferred  from  Kep- 
ler's law  of  areas,  for  it  is  a  well  known  property  of  the  ellipse, 
that  the  major  axis  is  the  only  line  drawn  through  the  focus  that 


LONGITUDE  AND  EPOCH  OF  THE  PERIGEE.  113 

divides  the  ellipse  into  equal  parts,  and,  by  the  law  in  question, 
equal  areas  correspond  to  equal  times. 

156.  Progressive  Motion  of  the  Perigee.  By  a  compari- 
son of  the  results  of  observations  made  at  distant  epochs,  it  is 
discovered  that  the  longitude  of  the  perigee  is  continually  inereas 
ing  at  a  mean  rate  of  6L".7  per  year.  As  the  equinox  retro- 
grades 50". 2  in  a  year,  the  perigee  must  then  have  a  direct  angu- 
lar motion  of  11  ".5  per  year. 

It  will  be  seen  that  as  a  consequence  the  interval  between  the§ 
times  of  the  sun's  passage  through  the  apogee  and  perigee,  is  not,' 
strictly  speaking,  half  a  sidereal  year,  but  exceeds  this  period  by 
the  interval  of  time  employed  by  the  sun  in  moving  through  an 
arc  of  5".7,  the  sidereal  motion  of  the  apogee  and  perigee  in  half 
a  year. 

According  to  the  most  exact  determinations,  the  mean  longi- 
tude of  the  perigee  of  the  sun's  orbit  at  the  beginning  of  the 
year  1800,  was  279°  29'  56".  It  is  now  280f°. 

157.  The  Heliocentric  Longitude  of  the  Perihelion  of 
the  Uarth's  Orbit,  is  equal  to  the  geocentric  longitude  of  the 
perigee  of  the  sun's  apparent  orbit  minus  180°.  For,  let  AEP 
(Fig.  49,  p.  103)  be  the  earth's  orbit,  and  PV  the  direction  of  the 
vernal  equinox.  When  the  earth  is  in  its  perihelion,  P,  the  sun 
is  in  its  perigee,  S,  and  we  have  the  heliocentric  longitude  of  the 
perihelion,  YSP  =  YPL  =  angle  abc  — 180°  =  geocentric  longi- 
tude of  the  sun's  perigee  — 180°.  It  is  plain  that  the  same 
relation  subsists  between  the  heliocentric  longitude  of  the  earth 
and  the  geocentric  longitude  of  the  sun  in  every  other  position 
of  the  earth  in  its  orbit. 

15§.  The  mean  Longitude  of  the  Sun,  at  any  assumed 
epoch,  may  be  obtained  by  means  of  the  mean  motion  in  longi- 
tude (116),  the  epoch  and  mean  longitude  of  the  perigee  of  the 
sun's  orbit  having  once  been  found. 


DETERMINATION  OF  THE  ELEMENTS  OF  THE  MOON'S 
ORBIT. 

LONGITUDE    OF  THE  NODE. 

159.  In  order  to  obtain  the  longitude  of  the  moon's  ascending 
node,  we  have  only  to  find  the  longitude  of  the  moon  at  the  time 
its  latitude  is  zero,  and  the  moon  is  passing  from  the  south  to 
the  north  side  of  the  ecliptic.  This  may  be  deduced  from  the 
longitudes  and  latitudes  of  the  moon,  derived  from  observed 
right  ascensions  and  declinations  (56) ;  by  methods  precisely  ana- 
logous to  those  by  which  the  right  ascension  of  the  sun,  at  the 
time  its  declination  is  zero,  and  it  is  passing  from  the  south  to 
the  north  side  of  the  equator,  or  the  position  of  the  vernal  equi- 
nox, is  ascertained  (113). 

8 


MOTION   OF  THE  MOON  IN  SPACE. 


INCLINATION  OF  THE  ORBIT. 

160.  Among  the  latitudes  computed  from  the  moon's  observed 
right  ascensions  and  declinations,  the  greatest  measures  the  incli- 
nation of  the  orbit.     It  is  found  to  be  about  5° ;  sometimes  a 
little  greater,  and  at  other  times  a  little  less. 

MEAN  MOTION. 

161.  Tropical  Revolution.     With  the  longitudes  of  the 
moon,  found  from  day  to  day,  it  is  easy  to  obtain  the  interval 
from  the  time  at  which  the  moon  has  any  given  longitude  till  it 
returns  to  the  same  longitude  again.     This  interval  is  called  a 
Tropical  Revolution  of  the  moon.     It  is  found  to  be  subject  to 
considerable  periodical  variations,  and  thus  one  observed  tropical 
revolution  may  differ  materially  from  the  mean  period.     In  order 
to  obtain  the  mean  tropical  revolution,  we  must  compare  two 
longitudes  found  at  distant  epochs.     Their  difference  augmented 
by  the  product  of  860°  by  the  number  of  revolutions  performed 
in  the  interval  of  the  epochs,  will  be  the  mean  motion  in  longi- 
tude in  the  interval ;  from  which  the  mean  motion  in  100  years, 
or  36,525  days,  called  the  Secular  motion,  may  be  obtained  by 
simple  proportion.     The  secular  motion  being  once  known,  it  is 
easy  to  deduce  from  it  the  period  in  which  the  motion  is  360°, 
which  is  the  mean  tropical  revolution. 


It  should  be  observed,  however,  that  to  find  the  precise  mean  secular  motion  in 
longitude,  it  is  necessary  to  compare  the  mean  longitudes  instead  of  the  true. 
Now,  the  true  longitude  of  the  moon  at  any  tune  having  been  found,  the  mean 
longitude  at  the  same  time  is  derived  from  it  by  correcting  for  the  equation  of  the 
centre  and  certain  other  periodical  inequalities  of  longitude  hereafter  to  be  noticed. 
But  this  cannot  be  done,  even  approximately,  until  the  theory  of  the  moon's  motions 
is  known  with  more  or  less  accuracy. 

The  longitude  of  the  moon,  at  certain  epochs,  may  be  very 
conveniently  deduced  from  observations  upon  lunar  eclipses.  For, 
the  time  of  the  middle  of  the  eclipse  is  very  near  the  time  of 
opposition,  when  the  longitude  of  the  moon  differs  180°  from 
that  of  the  sun,  and  the  longitude  of  the  sun  results  from  the 
known  theory  of  its  motion.  The  recorded  observations  of  the 
ancients  upon  the  times  of  the  occurrence  of  eclipses,  are  the  only 
observations  that  can  now  be  made  use  of  for  the  direct  determi- 
nation of  the  longitude  of  the  moon  at  an  ancient  epoch. 

162.  Mean  Daily  Motion  in  Longitude.  The  mean  tropi- 
cal revolution  of  the  moon  is  found  to  be 

•     27.321582d.  or  27d.  7h.  43m.  4.7s.  (5s.  nearly). 
Hence,  27.321582d.  :  Id. : :  360°  :  13M7639  =  13°  10'  35".0  =: 
moon's  mean  daily  motion  in  longitude. 


LONGITUDE  OF  THE  PERIGEE. 


115 


163.  Sidereal  Revolution.  Since  the  equinox  has  a  retro- 
grade motion,  the  sidereal  revolution  of  the  moon  roust  exceed 
the  tropical  revolution,  as  the  sidereal  year  exceeds  the  tropical 
year.  The  excess  will  be  equal  to  the  time  employed  by  the 
moon  in  describing  the  arc  of  precession  answering  to  a  revolution 
of  the  moon.  Thus, 

365.25d.  :  50".2  : :  27.3d. :  3".752  =  the  arc  of  precession, 
and  13°.17G  :  Id.  : :  3".752  :  6.8s.  =  excess. 

Wherefore,  the  mean  sidereal  revolution  of  the  moon  is  27d.  7h. 
43m.  11.5s. 

164.  Secular  Acceleration  of  Moon's  Ulotion.  It  has  been 
found,  by  determining  the  moon's  mean  rate  of  motion  for  periods  of  various  lengths, 
that  it  is  subject  to  a  continual  slow  acceleration.  This  acceleration  will  not,  how- 
ever, be  indefinitely  progressive;  Laplace  investigated  its  physical  cause,  and 
showed,  from  the  principles  of  Physical  Astronomy,  that  it  is  really  a  periodical 
inequality  in  the  moon's  mean  motion,  which  requires  an  immense  length  of  time 
to  go  through  its  different  values. 

The  mean  motion  given  in  Art.  162  answers  to  the  commencement  of  the  present 
century. 


LONGITUDE   OF  THE   PERIGEE,   ECCENTRICITY,   AND  SEMI-MAJOR 

AXIS. 

165.  The  methods  of  determining  these  elements  of  the 
moon's  orbit  are  similar  to  those  by  which  the  corresponding 
elements  of  the  sun's  orbit  are  found. 

166.  Orbit  JLongitudes.     The  only  essential  difference  in  the  methods 

adopted,  is  that  in  place  of  the  longitudes  of  the 
sun,  which  are  laid  off  in  the  plane  of  the  eclip- 
tic, hi  the  case  of  the  moon  corresponding  an- 
gles are  laid  off  hi  the  plane  of  its  orbit.  These 
angles  are  reckoned  from  a  line  drawn  making 
an  angle  with  the  line  of  nodes  equal  to  the 
longitude  of  the  ascending  node,  and  are  called 
Orbit  Longitudes.  The  orbit  longitude  is  equal 
to  the  moon's  angular  distance  from  the  ascend- 
ing node  plus  the  longitude  of  the  ascending 
node.  Thus,  let  TOO  (Fig.  53)  represent  the 
plane  of  the  ecliptic,  and  VNM  a  portion  of  the 
moon's  orbit;  N  being  the  ascending  node; 
also  let  EV  be  the  direction  of  the  vernal  equi- 
nox, and  let  EV  be  drawn  in  the  plane  of  the 
moon's  orbit,  making  an  angle  V'EN  with  the 
line  of  the  nodes  equal  to  VBN,  the  longitude 
of  the  ascending  node  N.  The  orbit  longitudes 
lie  in  the  plane  of  the  moon's  orbit,  and  are 
estimated  from  this  line,  while  the  ecliptic  lon- 
gitudes lie  in  the  plane  of  the  ecliptic,  and  are 
estimated  from  the  line  EV.  Thus,  V'EM,  or 
its  measure  V'NM,  is  the  orbit  longitude  of  the 
jnoon  hi  the  position  M;  and  YEm  is  the  eclip- 
tic longitude ;  that  is,  the  longitude  as  it  has 
been  hitherto  considered.  V'NM  =  VN  +  NM  =  YN"  +NM  •  that  is,  orbit  long. 
=  long,  of  %  +  3's  distance  from  £. 


FIG.  53. 


116  MOTIONS  OF  THE  PLANETS  IN  SPACE. 

The  orbit  longitudes  are  calculated  from  the  ecliptic  longitudes ;  these  being 
derived  from  observed  right  ascensions  and  declinations. 

167.  The  ecliptic  longitude  of  the  moon  at  any  time  being  given,  to  find  the  orbu 
longitude.    As  we  may  suppose  the  longitude  of  the  node  to  be  given  (159),  the 
equatiou  of  the  preceding  article  will  make  known  the  orbit  longitude  so  soon  as 
MN,  the  moon's  distance  from  the  node,  becomes  known :  now,  by  Napier's  first 
rule  we  have 

cos  MNm,  =  cot  NM  tan  Nra ; 
or,  cot  NM  =  cos  MNw  cot  Nm. 

Nm  =  ecliptic  long.  — long,  of  node ;  and  MNw  =  inclination  of  orbit. 

168.  The  Horizontal  Parallax  of  tlie  illooii,  like  almost 
every   other    astronomical    element,   is    subject   to    periodical 
changes  of  value.     It  varies  not  only  during  one  revolution, 
but  also  from  one  revolution  to  another.     The  fixed  and  mean 
parallax,  about  which  the  true  parallax  may  be  conceived  to 
oscillate,  answers  to  the  mean  distance,  that  is,  the  distance  about 
which  the  true  distance  varies  periodically,  and  is  called  the 
Constant  of  the  Parallax.     It  is,  for  the  equatorial  radius  of  the 
earth,  57'  2".7,  from  which  we  find  by  equation  (32)  the  mean 
distance  of  the  moon  from  the  earth  to  be  238,824-  miles. 

169.  The  Eccentricity  of  the  moon's  orbit  is  more  than 
three  times  as  great  as  that  of  the  sun's  apparent  orbit.     Its  great- 
est equation  exceeds  6°  (154). 


MEAN   LONGITUDE   AT  AN  ASSIGNED  EPOCH. 

17O.  We  have  already  explained  (161)  the  principle  of  the 
determination  of  the  mean  longitude  of  the  moon  from  an  ob- 
served true  longitude.  Now,  when  the  mean  longitude  at  any 
one  epoch  whatever  becomes  known,  the  mean  longitude  at  any 
assigned  epoch  is  easily  deduced  from  it  by  means  of  the  mean 
motion  in  longitude. 


DETERMINATION  OF  THE  ELEMENTS  OF  A  PLANET'S  ORBIT. 

171.  Heliocentric  Longitude  and  Radius-Vector  of 
the  Earth.  The  methods  of  determining  the  elements  of  the 
planetary  orbits,  suppose  the  possibility  of  finding  the  heliocen- 
tric longitude  and  radius- vector  of  the  earth  for  any  given 
time.  Now,  the  elements  of  the  earth's  orbit  having  been  found 
by  the  processes  heretofore  detailed,  the  longitude  may  be  com- 
puted by  means  of  Kepler's  first  law ;  and  the  radius- vector  from 
the  polar  equation  of  the  orbit,  as  given  in  treatises  on  Analyti- 
cal Geometry.  The  manner  of  effecting  such  computation  will 
be  considered  hereafter ;  at  present  the  possibility  of  effecting  it 
will  be  taken  for  granted. 


LONGITUDE   OF  THE  ASCENDING  NODE. 


117 


HELIOCENTRIC  LONGITUDE  OF  THE  ASCENDING  NODE. 

172.  First  Method.  "When  the  planet  is  in  either  of  its  nodes,  its  lati- 
tude is  zero.  It  follows,  therefore,  that  the  longitude  of  the  planet  at  the  time  its 
latitude  is  zero,  is  the  geocentric  longitude  of  the  node  at  the  time  the  planet  is 
passing  through  it.  Now,  if  the  right  ascension  and  declination  of  the  planet  be 
observed  from  day  to  day,  about  the  time  it 
is  passing  from  one  side  of  the  ecliptic  to  the 
other,  and  converted  into  longitude  and  lati- 
tude, the  time  at  which  the  latitude  is  zero, 
and  the  longitude  at  that  time,  may  be  ob- 
tained by  a  proportion.  When  the  planet  is 
again  in  the  same  node,  the  geocentric  longi- 
tude of  the  node  may  again  be  found  in  the 
same  manner  as  before.  On  account  of  the 
different  position  of  the  earth  in  its  orbit,  this 
longitude  will  differ  from  the  former. 

Now,  if  two  geocentric  longitudes  of  the  same 
node  be  found,  its  heliocentric  longitude  may  be 
computed.  Let  S  (Fig.  64)  be  the  sun,  N 
the  node,  and  E  one  of  the  positions  of  the 
earth  for  which  the  geocentric  longitude  of 
the  node  (YEN)  is  known.  Denote  this 
angle  by  G-,  the  sun's  longitude  YES  by  S, 
and  the  radius- vector  SE  by  r.  Also,  let  E' 
be  the  other  position  of  the  earth,  and  de- 
note the  corresponding  quantities  for  this 
position,  YE'N,  VE'S,  and  SE',  respectively, 
by  G',  S',  and  r*.  Let  the  radius-vector  of 
the  planet  when  in  its  node,  or  SN  =  V ;  and  the  heliocentric  longitude  of  the 
node,  or  VSN  =  X.  The  triangle  SNE  gives 

sin  SNE  :  sin  SEN.  : :  SE  :  SN; 

but  SEN  =  VES  —  YEN  =  S  —  G, 

and  SNE  =  VAN  —  VSN  =  VEN  —  VSN  =  G— X; 


FIG.  54. 


hence, 

or, 

In  like  manner, 

Dividing, 


or     r  sm  S  "" 
' 


sin  (G  —  X)  :  sin  (S  —  G)  :  :  r  :  V, 
r  sin  (S  —  G)  =  Y  sin  (G  —  X).  .  .  .(37). 
r'  sin  (S'  —  G')  =  Y  sin  (G;  —  X). 
r  sin  (S  —  G)  __  sin  (G  —  X) 
r'  sin  (S'  ^G7)  ~~  sin  (G'-^X)' 
sm  0-  c08  X—  sin  X  cos  G  _  sin  G  —  cos  G  tan  X 


whence, 


(S'—  G7)      sin  G'  cos  X  —  sin  X  cos  G'      sin  G'  —  cos  G'  tan  X 


_  r  sin  (S  —  G)  sin  G'  —  r'  sin  (S'  —  G')  sin  G 
~  rsin  (S"~G)cosG'  —  r'  sin  (S'  —  G')  cos~G 


Equation  (37)  gives 


Y  =  LZ       ....  (39). 
sin  (G  —  X) 


173.  Second  method.  The  longitude  of  the  node  may  also 
be  found  approximately  from  observations  made  upon  the  planet 
at  the  time  of  conjunction  or  opposition.  It  will  happen  in  pro- 
cess of  time  that  some  of  the  conjunctions  and  oppositions  will 
occur  when  the  planet  is  near  one  of  its  nodes  ;  the  observed 
longitude  of  the  sun  at  this  conjunction  or  opposition,  will  either 
be  approximately  the  heliocentric  longitude  of  the  node  in  ques- 
tion, or  will  differ  180°  from  it.  This  will  be  seen  on  inspecting 


113 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


Fig.  55.  If  at  a  certain  time  the  earth  should  be  at  E,  crossing 
the  line  of  nodes,  and  the  planet  in  conjunction,  it  will  be  in  the 
node  N,  and  YES,  the  longitude  of  the  sun,  will  be  equal  to  YSN, 
the  heliocentric  longitude  of  the  node.  If  the  earth  should  be 


FIG.  55. 

at  E"  and  the  planet  in  opposition,  the  longitude  of  the  sun 
would  be  VE"S  =:  VE"N  +  180°  =  YSN  +  180°  =  hel.  long, 
of  node  4- 180°. 

If  the  daily  variations  of  the  latitude  of  the  planet  should  be 
observed  about  the  time  of  the  supposed  conjunction  or  opposi- 
tion near  the  node,  the  time  when  the  latitude  becomes  zero,  or 
the  planet  is  in  its  node,  could  approximately  be  calculated  by 
simple  proportion  ;  and  then  so  soon  as  the  rate  of  the  angular 
motion  about  the  sun  becomes  known  (176)  the  longitude  of  the 
node  could  be  more  accurately  determined. 

INCLINATION  OF  THE   ORBIT. 

174.  The  longitude  of  the  node  having  been  found  by  the 
preceding,  or  some  other  method,  compute  the  day  on  which  the 
sun's  longitude  will  be  the  same  or  nearly  the  same :  the  earth 
will  then  be  on  the  line  of  the  nodes.  Observe  on  that  day  the 
planet's  right  ascension  and  declination,  and  deduce  the  geocen- 
tric longitude  and  latitude.  Let  ~ENp  (Fig.  55)  be  the  plane  of 
the  ecliptic,  V  the  vernal  equinox,  S  the  sun,  N  the  node,  E  the 
earth  on  the  line  of  the  nodes,  and  P  the  planet  as  referred  to 
the  celestial  sphere,  from  the  earth.  Let  /L  denote  the  geocentric 
latitude  PEp  ;  E  the  arc  Np  =  Vp  —  YN  =  geo.  long,  of  planet 
— -  long,  of  node ;  and  I  the  inclination  PNp.  The  right-angled 
triangle  PNp  gives 


PERIODIC  TIME.  119 

sin  "Np  =  tan  Pp  cot  P^p  =  tan  /t  cot  I  ; 


nence,          cot  I  =  5E,  and  tan  I  =     2_    ; 
tan  V  sin  E 

or,  tan  inclination  = 


sin  (long.  —  long,  of  node) 

It  will  be  understood,  that  to  obtain  an  exact  result,  we  must  compute  the  pro- 
case  time  of  day  at  which  the  longitude  of  the  sun  is  the  same  as  that  of  the 
node,  and  then,  by  means  of  their  observed  daily  variations,  correct  the  longitude 
and  latitude  of  the  planet  for  the  variations  hi  the  interval  between  the  time  thus 
ascertained  and  the  tune  of  the  observation  above  mentioned. 


PERIODIC  TIME. 

175.  The  interval  from  the  time  the  planet  is  in  one  of  its 
nodes  till  its  return  to  the  same,  gives  the  periodic  time  or  side- 
real revolution. 

Another  aiid  more  accurate  method  is  to  observe  the  length 
of  a  synodic  revolution  and  compute  the  periodic  time  from 
this.  If  we  compare  the  time  of  a  conjunction  which  has 
been  observed  in  modern  times,  with  that  of  a  conjunction  ob- 
served by  the  earlier  astronomers,  and  divide  the  interval  between 
them  by  the  number  of  synodic  revolutions  contained  in  it,  we 
shall  have  the  mean  synodic  revolution  with  great  exactness ; 
from  which  the  mean  periodic  time  may  be  deduced,  as  will  be 
shown  hereafter. 

176.  Mean  Daily  Motion.  The  periodic  time  being  known, 
the  mean  daily  motion  around  the  sun  may  be  found  by  dividing 
360°  by  the  periodic  time  expressed  in  days  and  parts  of  a  day. 

TO  FIND  THE    HELIOCENTRIC    LONGITUDE  AND  LATITUDE,   AND 
THE  RADIUS-VECTOR,   FOR  A  GIVEN  TIME. 

177.  General  Problem.  The  earth  being  in  constant  motion 
in  its  orbit,  and  being  thus  at  different  times  very  differently 
situated  with  regard  to  the  other  planets,  as  well  in  respect  to 
distance  as  direction,  it  is  necessary  for  the  purpose  of  compar- 
ing the  observations  made  upon  these  bodies  with  each  other,  to 
refer  them  all  to  one  common  point  of  observation.     As  the  sun 
is  the  fixed  centre  about  which  the  revolutions  of  the  planets  are 
performed,  it  is  the  point  best  suited  to  this  purpose,  and  accor- 
dingly it  is  to  the  sun  that  the  observations  are  in  reality  refer- 
red.    The  reduction  of  observations  from  the  earth  to  the  sun, 
as  it  is  actually  performed,  consists  in  the  deduction  of  the  helio- 
centric longitude  and  latitude  from  the  geocentric  longitude  and 
latitude ;  these  being  calculated  from  the  observed  right  ascen- 
sion and  declination. 

The  requisite  formulae  for  effecting  this  reduction  are  investi- 
gated in  the  Appendix. 


120 


MOTIONS  OF  THE  PLANETS  IN  SPACE. 


ITS.  Special  Ca§es.  The  heliocentric  longitude,  or  radius- 
vector  of  a  planet,  may  be  more  readily  obtained  if  the  observa- 
tions be  made  upon  it  when  it  is  in  certain  favorable  positions. 
€a§e  I.    When  the  planet  is  in  conjunction  or  opposition,  its  helio- 
centric longitude  will  then,  either  be 
equal  to  the  geocentric  longitude,  or 
differ  180°  from  it. 

When  the  heliocentric  longitude 
is  thus  found,  the  latitude  for  the 
same  time  may  be  obtained  by  solv- 
ing the  triangle  PNjp  (Fig.  56). 
For,  by  Napier's  first  rule, 

sin  ~Np  =  cot  PNp  tan  Pp, 
or  tan  Pp  =  sin  ~Np  tan  PNp ; 

where  Pp  is  the  latitude  sought, 
PNp  the  known  inclination  of  the 
orbit,  and  Np  =  YNp  —  YN  = 
long,  of  planet  —  long,  of  node,  both 
of  which  may  be  considered  as 
known. 

The  radius-vector  may  be  computed  for  the  same  time  from  the 
triangle  ESP ;  for  the  side  SE,  the  radius-vector  of  the  earth,  is 
known,  as  well  as  the  angle  SEP,  the  geocentric  latitude  of  the 
planet,  and  the  angle  ESP  =  180°  —  PSp  =  180°  —  heliocen- 
tric latitude. 

Ca§e  II.  When  an  inferior  planet  is  at  its  'maximum  elongation 
from  the  sun.  Tbe  radius- vector  of  either  of  the  inferior  planets 
at  the  time  of  maximum  elongation,  or  greatest  angular  distance 
from  the  sun,  may  be  approximately  deduced  from  the  amount 
of  the  greatest  elongation  determined  from  observation.  The 
elongation  which  obtains  at  any  time,  may  be  found  by  ascer- 
taining from  instrumental  observations  the  places  of  the  planet 
and  sun  in  the  heavens,  and  connecting  these  by  an  arc  of  a  great 
circle,  and  with  the  pole  by  other  arcs.  In  the  triangle  PSp 
(Fig.  57)  thus  formed,  there  will  be  known  the  two  polar  dis- 


FIG.  56. 


FIG.  57. 


tances  PS  and  Pp,  which  are  the  complements  of  the  observed 
declinations,  and  the  angle  SPp  the  difference  of  their  observed 
right  ascensions,  from  which  the  angular  distance  Sp  between  the 
two  bodies  may  be  calculated.  The  maximum  elongation  being 


LONGITUDE   OF   THE  PERIHELION. 


121 


then  supposed  to  be  known,  let  NPP'  (Fig.  58)  represent 
the  orbit  of  the  inferior  pla- 
net. The  line  EP  drawn 
from  the  earth  to  the  planet 
will,  at  the  time  of  maxi- 
mum elongation,  be  perpen- 
dicular to  SP,  the  radius- 
vector  of  the  planet;  and 
thus  we  shall  have  in  the 
right-angled  triangle  EPS, 
the  line  ES,  and  the  angle 
SEP,  from  which  the  radius- 
vector  SP  may  be  computed. 

As  the  earth  and  planet 
are  in  motion,  the  greatest 
elongation  will  occur  at  dif- 
ferent points  of  the  planet's 
orbit,  and  therefore  we  may  find  by  the  foregoing  process  different 
radius- vectors. 

179.  The  Orbit  Longitude  of  a  Planet  may  be  derived 
from  the  ecliptic  longitude  in  the  same  manner  that  the  orbit 
longitude  of  the  moon  is  calculated  from  its  ecliptic  longitude 
(160).  The  orbit  longitude  and  radius- vector,  when  found  for  a 
given  time,  ascertain  the  position  of  the  planet  in  the  plane  of 
its  orbit  at  that  time. 


LONGITUDE  OF  THE  PERIHELION,  ECCENTRICITY,  AND 
SEMI-MAJOR  AXIS. 

t§O.     These  elements  may  be  calculated  from  the  heliocentric 
orbit  longitude  and  radius-vector,  found  for  three  different  times. 

Let  SP,  SP',  SP"  (Fig.  59),  be  the  three  given  radius- vectors; 
V'SP,  V'SP',  V'SP",  the 
three  given  longitudes ;  and 
AB  the  line  of  apsides  of 
the  planet's  orbit.  Let  the 
angles  PSP',  PSP",  which 
are  known,  be  represented 
by  m  and  w,  and  the  angle 
BSP,  which  is  unknown,  by 
x  ;  and  let  the  radius- vectors 
SP,  SP',  SP",  be  denoted 
by  v,  v',  v",  the  semi-major 
axis  AC  by  a,  and  the  eccentricity  by  e.  Then  the  three 
unknown  quantities  which  are  to  be  determined  are  a,  e,  and  the 
angle  x ;  and  the  general  polar  equation  of  the  ellipse  furnishes 
for  their  determination  the  three  equations, 


Pro.  59. 


122  MOTIONS  OF  THE  PLANETS  IN  SPACE. 


1  -f-  e  cos  x1  1  -f-  e  cos  (x  -r  m)'  1  +  e  cos  (x  +  w)' 

The  process  of  solution  is  given  in  the  Appendix.  When  x  has 
been  found,  by  subtracting  it  from  V'SP  we  obtain  V'SB,  the 
longitude  of  the  perihelion. 

181.  Other  Methods  of  Determining  the  Semi-major 
Axis.  The  semi-major  axis,  or  mean  distance  from  the  sun,  may 
also  be  had  by  taking  the  mean  of  a  great  number  of  values  of 
the  radius-vector  found  for  every  variety  of  position  of  the 
planet  in  its  orbit  (178). 

Now  that  Kepler's  third  law  has  been  established  by  investi- 
gations in  Physical  Astronomy,  it  furnishes  the  most  accurate 
method  of  finding  the  mean  distance  of  a  planet  from  the  sun. 
Thus  let  P  denote  the  periodic  time  of  a  planet,  and  a  its  mean 
distance  from  the  sun  ;  then  the  length  of  the  sidereal  year  being 
365.256359  days  (120), 

(365.256359d.)a :  P3 : :  I3:  a3; 
whence,  a  =  [TSSFSSL^GEST 


182.  'Longitude  of  Perihelion,  and  Eccentricity,  by 
Approximate  Methods.  If  a  great  number  of  values  of  the 
radius-vector,  in  a  great  variety  of  positions  of  the  planet  in  its 
orbit,  be  found  by  the  method  explained  in  Art.  178,  the  longi- 
tude of  the  planet  at  the  time  when  its  calculated  radius-vector 
is  the  least,  will  be  approximately  the  longitude  of  the  perihelion  ; 
or,  if  it  chances  that  among  the  calculated  radius-  vectors  there 
are  two  equal  to  each  other,  the  position  of  the  line  of  apsides 
may  be  found  by  bisecting  the  angle  included  between  these. 

The  ratio  of  the  difference  between  the  greatest  and  least  cal- 
culated radii  to  the  mean  of  the  whole,  will  be  the  approximate 
value  of  the  eccentricity. 

EPOCH  OF  THE  PERIHELION  PASSAGE. 

1§3.  From  several  observations  upon  the  planet  about  the 
time  it  has  the  same  longitude  as  the  perihelion,  the  correct  time 
of  its  being  at  the  perihelion  may  be  easily  determined  by  pro- 
portion. 

The  Mean  Longitude  at  an  assigned  epoch  is  obtained  on  the 
same  principles  as  the  mean  longitude  of  the  sun  or  moon  (158, 
170). 

REMARKS. 

1§4.  The  foregoing  methods  of  determining  the  elements  of  a 
planet's  orbit  suppose  observations  to  be  made  at  two  or  more 
successive  returns  of  the  planet  to  its  node  ;  but  it  is  not  necessary 
to  wait  for  the  passage  of  a  planet  through  its  node.  Soon  after 


TRUE  AND  MEAN  ELEMENTS.  123 

the  planet  Uranus  was  discovered  by  Sir  William  Herschel,  La- 
place contrived  methods  by  which  the  elements  of  its  elliptic 
orbit  were  determined  from  four  observations  within  little  more 
than  a  year  from  its  first  discovery  by  Herschel.  After  the  dis- 
covery of  Ceres,  Gauss  invented  another  general  method  of  cal- 
culating the  orbit  of  a  planet  from  three  observations,  and  applied 
it  to  the  determination  of  the  orbit  of  Ceres,  and  subsequently  to 
the  determination  of  the  orbits  of  Pallas,  Juno,  and  Vesta.  This 
method  can  be  more  readily  employed  in  practice  than  that  of 
Laplace,  or  than  any  of  the  solutions  which  other  mathematicians 
have  given  of  the  same  problem,  and  is  now  generally  used  in 
computing  the  orbit  of  a  newly  discovered  planet.  <f 

TRUE  AND  MEAN  ELEMENTS. 

185.  True  dements  and  their  variations.    The  elements  of  the  planetary  orbits, 
obtained  by  the  foregoing  processes,  are  the  true  elements  at  the  periods  when 
the  observations  are  made.     Upon  determining  them  at  different  periods,  it  appears 
that  they  are  subject  to  minute  variations.     A  comparison  of  the  values  found  at 
various  distant  epochs  shows  that  they  are  slowly  changing  from  century  to  cen- 
tury, and  that  the  changes  experienced  during  equal  long  periods  of  time  are  very 
nearly  the  same.     The  amount  of  the  variation  of  an  element  iu  a  period  of  100 
years  is  called  its  Secular  Variation.     Upon  reducing  the  elements,  found  at  differ- 
ent times,  to  the  same  epoch,  by  allowing  for  the  proportional  parts  of  the  secular 
variations,  the  different  results  for  each  element  are  found  to  differ  slightly  from 
each  other,  which  shows  that  the  elements  are  also  subject  to  slight  periodical 
variations.     These  variations  being  very  minute,  the  true  elements  can  never  differ 
much  from  the  mean,  or  those  from  which  they  deviate  periodically  and  equally  on 
both  sides. 

186.  Mean  Elements  and  their  Secular  Variations.    The  mean  elements  at  an 
assigned  epoch  may  be  had  by  finding  the  true  elements  at  various  tunes,  and  re- 
ducing them  to  the  given  epoch,  by  making  allowance  for  the  proportional  parts 
of  the  secular  variations,  and  then  taking  for  each  element  the  mean  of  all  the  par- 
ticular values  obtained  for  it. 

A  comparison  of  the  mean  values  of  the  same  element,  found  at  distant  epochs, 
makes  known  the  variation  of  its  mean  value  in  the  interval  between  them,  from 
which  the  secular  variation  may  be  deduced  by  simple  proportion. 

187.  Variations  of  Elements  of  Moon's  Orbit.    The  elements  of  the  moon's  orbit 
are  also  subject  to  continual  variations.    These  are,  for  the  most  part,  periodic, 
and  are  far  greater  than  the  variations  of  the  corresponding  elements  of  a  planet's 
orbit.     It  will  be  seen,  then,  that  hi  determining  the  mean  elements,  a  much  greater 
number  of  observations  will  be  required  than  in  the  case  of  a  planetary  orbit. 
The  mean  node  and  perigee  have  a  rapid  and  nearly  uniform  motion.     Their 
motions,  in  connection  with  the  mean  motion  of  revolution  of  the  moon,  are 
subject  to  minute  secular  variations.     The  mean  eccentricity,  and  inclination  of  the 
orbit,  are  constant. 

188.  Verifications.    The  mean  elements  which  have  been  derived  as  above, 
directly  from  observation,  have  subsequently  been  verified  and  corrected  by  com- 
paring the  computed  with  the  observed  places  of  the  planet ;  and  for  this  purpose 
many  thousands  of  observations  have  been  made. 

1§9.  Tables  II.  and  III.  contain  the  elements  of  the  orbits 
of  the  principal  planets,  and  of  the  moon's  orbit,  together  with 
their  secular  variations  for  the  beginning  of  the  year  1850. 
Table  II.  (a)  contains  the  mean  distances,  sidereal  revolutions, 
and  eccentricities  of  the  orbits  of  the  planetoids. 


124  MOTIONS  OF  THE   PLANETS  IN  SPACE. 

If  an  element  be  desired  for  anytime  different  from  the  epoch 
of  the  table,  we  have  only  to  allow  for  the  proportional  part  of 
the  secular  variation,  in  the  interval  between  the  given  time  and 
the  epoch  of  the  table. 

190.  Secular  Variations.     It  will  be  seen   on  inspecting 
Table  II.,  that  the  mean  distances  of  the  planets  from  the  sun,  or 
the  semi-major  axes  of  their  orbits,  are  the  only  elements  that 
are  invariable.     The  rest  are  subject  to  minute  secular  variations. 
The  nodes  have  all  retrograde  motions.     The  perihelia,  on  the 
contrary,  have  direct  motions,  with  the  single  exception  of  the 
perihelion  of  the  orbit  of  Yenus,  which  has  a  retrograde  motion. 
The  eccentricities  of  some  of  the  orbits  are  increasing;  of  others, 
diminishing.     That  of  the  earth's  orbit  is  diminishing. 

The  node  of  the  moon's  orbit  has  a  retrograde  motion,  and  the 
perihelion  a  direct  motion.  The  former  accomplishes  a  tropical 
revolution  in  about  18  years  and  224  days,  and  the  latter  in 
about  8  years  and  309  (lays.  The  mean  motion  of  the  node 
and  the  mean  motion  of  the  perigee  are  both  subject  to  a  slow 
secular  diminution. 

191.  Eccentricities  and  Inclinations.     It  will   be  seen 
also,  that  the  orbits  of  the  principal  planets  are  ellipses  of  small 
eccentricity,  or  which  differ  but  slightly  from  circles ;  and  that 
they  are  inclined  under  small  angles  to  the  plane  of  the  ecliptic. 
The  eccentricity  is  in  almost  every  instance  so  small  that,  if  a 
representation  of  the  orbit  were  accurately  delineated,  it  would 
not  differ  perceptibly  from  a  circle.     The  most  eccentric  orbits 
are  those  of  Mercury  and  Mars;  and  the  least  eccentric  those  of 
Venus,  Neptune,  and  the  earth.     The  eccentricity  of  Mercu^'s 
orbit  is  12  times  that  of  the  Dearth's,  of  Mars  6  times,  of  Venus 
less  than  J.     The  eccentricities  of  the  orbits  of  Jupiter,  Saturn, 
and  Uranus,   are   each   about  three   times  that  of  the  earth's 
orbit. 

The  orbit  of  Mercury  is  more  inclined  to.  the  ecliptic  than  the 
orbit  of  any  other  of  the  eight  principal  planets;  and  the  orbit 
of  Uranus  is  less  inclined  than  that  of  any  other  planet.  The 
inclination  of  the  latter  is  f°,  of  the  former  7°. 

The  orbits  of  the  planetoids  are  in  general  more  eccentric,  and 
more  inclined  to  the  plane  of  the  ecliptic  than  those  of  the  other 
planets.  The  inclination  of  the  orbit  of  Pallas  is  nearly  35°. 

192.  The  Mean  Distances  of  the  Planets  from  the  *im, 
expressed  in  miles,  are  in  round  numbers  as  follows :  Mercury 
35  millions,  Venus  66  millions,  the  earth  91  millions,  Mars  139 
millions,  Juno  244  millions,  Jupiter  475  millions,  Saturn  871 
millions,  Uranus  1752  millions,  Neptune  2,743  millions.     The 
range  of  distance  is  from  1  to  -77-J-.     The  distance  of  Neptune 
is  30  times  the  earth's  distance. 

193*  The  Approximate  Periods  of  Revolution  of  the 
planets  are:  of  Mercury  3  months,  Venus  7j  months,  Mars  1$ 


DIMENSIONS   OF  THE  SOLAR  SYSTEM.  125 

years,  Juno  4f  years,  Jupiter  a  little  less  than  12  years,  Saturn 
29-|-  years,  Uranus  84  years,  Neptune  164^  years.  The  periods 
and  mean  distances  are  more  exactly  given  in  Table  II.  (For  the 
planetoids,  see  Table  II.  (a)  ). 

194.  Bode's   L,aw.     A  remarkable  empirical   law,   called 
Bodes  Law  of  the  Distances,  was  announced  in  1772  by  Professor 
Bode,  of  Berlin,  as  connecting  the  distances  of  the  planets  from 
the  sun.     It  is  as  follows  :  If  we  take  the  numbers  0,  3,  6, 12,  24, 
48,  96,  192,  and  add  4  to  each  one  of  them,  so  as  to  obtain  4,  7, 
10,  16,  28,  52,  100, 196,  this  series  of  numbers  will  express  the 
order  of  distance  of    the   planets  from   the  sun.      This  law 
embodies  the  following  curious  relation  between  the  distances 
of  the  orbits  from  one  another,  viz. :  setting  out  from  Venus, 
the   distance   between   two   contiguous  orbits  increases  nearly 
in  a  duplicate  ratio  as  we  recede  from  the  sun  ;  that  is,  the  dis- 
tance from  the  orbit  of  the  earth  to  the  orbit  of  Mars,  is  twice  the 
distance  from  the  orbit  of  Venus  to  the  orbit  of  the  earth,  and 
one-half  the  distance  from  the  orbit  of  Mars  to  the  orbits  of  the 
planetoids. 

Previous  to  the  discovery  of  the  planetoids,  to  .complete  the 
above  law  a  planet  was  wanting  between  Mars  and  Jupiter.  It 
was,  on  this  account,  surmised  by  Bode  that  another  planet 
might  exist  between  these  two.  Instead  of  one  such  planet, 
however,  no  less  than  ninety-one  have  since  been  discovered, 
revolving  at  pretty  nearly  the  distance  from  the  sun  that 
Bode  had  derived  from  his  law  for  the  distance  of  the  sup- 
posed planet ;  some  at  a  little  greater,  and  others  at  a  little  less 
distance. 

Bode's  law,  though  it  holds  good  for  the  planets  in  general, 
fails  in  the  case  of  the  planet  Neptune;  the  error  for  this 
planet  being  more  than  one-fourth  the  whole  distance.  The 
error  is  one-twentieth  of  the  distance  for  Mars,  and  also  for 
the  planetoids.  For  Mercury,  Venus,  and  Saturn  it  is  about 
one-thirtieth.  For  Uranus  and  Jupiter  it  is  a  still  smaller 
fraction. 

195.  Dimensions  of  the  Solar  System.     A  better  idea  of 
the  dimensions  of  the  solar  system  than  is  conveyed  by  the  state- 
ment of  distances  above  given,  may  be  gained  by  reducing  its 
scale  sufficiently  to  bring  it  within  the  range  of  familiar  distances. 
Thus,  if  we  suppose  the  earth  to  be  represented  by  a  ball  only  one 
inch  in  diameter,  the  distance  of  Mercury  from  the  sun  will  be 
represented,  on  the  same  scale,  by  370  feet,  the  distance  of  Venus 
by  700  feet,  that  of  the  earth  by  960  feet,  that  of  Mars  by  1,500 
feet,  that  of  Juno  by  half  a  mile,  that  of  Jupiter  by  1  mile, 
that  of  Saturn  by  If  miles,  that  of  Uranus  by  3£  miles,  and  that 
of  Neptune  by  5£  miles.     On  the  same  scale,  the  distance  of 
the  moon  from  the  earth  would  be  only  2|  feet. 


126  DETERMINATION  OF  THE   PLACE  OF  A  PLANET. 


CHAPTER  X. 

DETERMINATION  OF  THE  PLACE  OF  A  PLANET,  OR  OF  THE  SUN 
OR  MOON,  FOR  A  GIVEN  TIME,  BY  THE  ELLIPTIC  THEORY.— 
VERIFICATION  OF  KEPLER'S  LAWS. 

PLACE  OF  A  PLANET  IN  ITS  ORBIT. 

196.  True  and  Mean  Anomaly.  The  angle  contained 
between  the  line  of  apsides  of  a  planet's  orbit  and  the  radius- 
vector,  as  reckoned  from  the  perihelion  towards  the  east,  is  called 
the  True  Anomaly.  Thus,  let  BPAP'  (Fig.  60)  represent  the 


FIG.  60. 


orbit,  B  the  perihelion,  and  P  the  position  of  the  planet ;  then 
BSP  is  its  true  anomaly.  The  angle  contained  between  the  line 
of  apsides  and  the  mean  place  of  the  planet,  also  reckoned  from 
the  perihelion  towards  the  east,  is  called  the  Mean  Anomaly. 
Thus,  let  M  be  the  mean  place  of  a  planet  at  the  time  P  is  its 
true  place,  and  BSM  will  be  its  mean  anomaly.  The  difference 
between  the  true  anomaly  BSP  and  the  mean  anomaly  BSM,  is 
the  angular  distance  MSP  between  the  true  and  mean  place  of 
the  planet,  or  the  equation  of  the  centre  (143). 

Describe  a  circle  Bp  A  on  the  line  of  apsides  as  a  diameter ; 
through  P  draw  ^PD  perpendicular  to  the  line  of  apsides,  and 
join  p  and  C ;  the  angle  BCp,  which  the  line  thus  deter- 
mined makes  with  the  line  of  apsides,  is  called  the  Eccentric 
Anomaly. 

The  corresponding  angles  appertaining  to  the  sun's  apparent 
orbit,  and  to  the  moon's  orbit,  have  received  the  same  appellations. 

19T.  Anomalistic  Revolution.     The  interval  between  two 


HELIOCENTRIC   PLACE   OF  A  PLANET.  127 

consecutive  returns  of  a  body  to  either  apsis  of  its  orbit,  is  called 
the  Anomalistic  Revolution.  The  anomalistic  revolution  of  the 
earth,  or  of  the  sun  in  its  apparent  orbit,  is  termed,  also,  the 
Anomalistic  Year. 

The  periodic  time,  or  the  mean  motion  of  a  body,  and  the 
motion  of  the  apsis  of  its  orbit,  being  known,  the  anomalistic 
revolution  may  be  easily  computed.  Let  m  =  the  sidereal 
motion  of  the  apsis  answering  to  the  periodic  time,  and  M  =  the 
mean  daily  motion  of  the  planet ;  then, 

M  :  Id.  :  :  m  :  x  —  diff.  of  anomalistic  rev.  and  periodic  time. 

When  the  epoch  of  any  one  passage  of  a  planet  through  its 
perihelion,  or  of  the  sun  or  moon  through  its  perigee,  has  been 
found,  we  may,  by  means  of  the  anomalistic  revolution,  deduce 
from  it  the  epoch  of  every  other  passage. 

The  length  of  the  anomalistic  year  exceeds  that  of  the  sidereal 
year  by  4m.  39s. 

19§.  Calculation  of  iTIcaii  Anomaly.  From  the  anoma- 
listic revolution,  and  the  epoch  of  the  last  passage  through  the 
perihelion  or  perigee  (as  the  case  may  be),  we  may  derive  the 
mean  anomaly  for  any  given  time.  Let  T  =  the  anomalistic 
revolution,  t  =  the  time  that  has  elapsed  since  the  last  passage 
through  the  perihelion  or  perigee,  and  A  ==  the  mean  anomaly  ; 
then, 

T:360°::*:  A  =  860° -^ . . . .  (42). 

199.    The  Place    of    a    Body  in   its    Elliptic   Orbit   is 

ascertained  by  finding  its  true  anomaly.  The  problem  which  has 
for  its  object  the  determination  of  the  true  anomaly  from  the 
mean,  was  first  resolved  by  Kepler,  and  is  called  Kepler's  Prob- 
lem. Another  and  more  convenient  method  of  obtaining  the 
true  anomaly,  is  to  compute  the  equation  of  the  centre  from  the 
mean  anomaly,  and  add  it  to  the  mean  anomaly,  or  subtract  it 
from  it,  according  to  the  position  of  the  body  in  its  orbit  (143). 
(See  Appendix,  Solution  of  Kepler's  Problem.) 


HELIOCENTRIC  PLACE  OF  A  PLANET. 

200.  The  Place  of  a  Planet  in  the  Plane  of  its  Orbit 

is  designated  by  its  orbit  longitude  (166)  and  radius-vector.  To 
find  the  orbit  longitude  we  have  the  equation  V'SP  =  V'SB  + 
BSP(seeFig.  60);  or, 

long.  =  long,  of  perihelion  +  true  anomaly. 
The  orbit  longitude  may  also  be  deduced  from  the  mean  lon- 
gitude, by  adding  or  subtracting  the  equation  of  the  centre  ;  for, 


128      DETERMINATION  OF  THE  PLACE  OF  A  PLANET. 

or,  true  long.  =  mean  long.  +  equa.  of  centre : 

also,  V'SP'  =  Y'SM'  —  M'SP', 
or,  true  long.  =  mean  long.  —  equa.  of  centre. 

The  radius-vector  results  from  the  polar  equation  of  the  ellip- 
tic orbit,  viz. : 

v=o(^-fl 

1  +  e  cos  x 

in  which  x  denotes  the  true  anomaly,  e  the  eccentricity,  and  a 
the  semi-major  axis. 
2OI.  To  find  tSie  Heliocentric  Longitude  and  Latitude, 

which  ascertain  the  position  of  the  planet  with  respect  to  the 
ecliptic,  the  triangle  NP/>  (Fig.  56,  p.  120)  gives 

sin  Pp  =  sin  NP  sin  PNp  ; 

or,  sin  lat.  =  sin  (orbit  long. — long,  of  node)  x  sin  (inclin.) . .  (44) ; 
and 

cos  PNp  —  tan  Np  cot  NP,  or  tan  Np  =  tan  NT  cos  PNjp, 
or, 

tan  (long.  —  long,  of  node)  =  tan  (orbit  long.  —  long,  of  node) 
X  cos  (inclination) ....  (45). 


GEOCENTRIC  PLACE  OF  A  PLANET. 

202.  The  theoretical  determination  of  the  place  of  a  planet,  as 
it  would  be  seen  from  the  centre  of  the  earth,  consists  in  deduc- 
ing its  geocentric  longitude  and  latitude,  and  its  distance  from 
the  earth,  from  its  heliocentric  longitude  and  latitude  and  radius- 
vector;  the  latter  having  been  calculated  by  the  methods  just 
explained.     (For  the  detail  of  the  solution  of  this  problem  see 
Appendix.) 

PLACES  OF  THE  SUN  AND  MOON. 

203.  The  place  of  the  sun,  as  seen  from  the  earth,  may  be 
easily  deduced  from  the  heliocentric  place  of  the  earth  ;  for  the 
longitude  of  the  sun  is  equal  to  the  heliocentric  longitude  of  the 
earth  plus  180°  (157),  and  the  radius-vector  of  the  earth's  orbit 
is  the  same  as  the  distance  of  the  sun  from  the  earth.     But  it  is 
more  convenient  to  regard  the  sun  as  describing  an  orbit  around 
the  earth,  and  compute  its  true  anomaly  (199) ;  and  thence  the 
longitude  and  radius- vector,  by  the  equation 

long.  =  true  anomaly  -f-  long,  of  perigee, 
and  the  polar  equation  of  the  orbit. 

204.  The  Orbit  Longitude  and  the  Radiu§- vector  of 
the  Moon,  are  found  by  the  same  process  as  the  longitude  and 
radius-vector  of  the  snn.     The  orbit  longitude  being  known. 


VERIFICATION  OF  KEPLER'S  LAWS.  129 

the  ecliptic  longitude  and  the  latitude  may  be  determined  bj  a 
process  precisely  similar  to  that  by  which  the  heliocentric  longi- 
tude and  latitude  of  a  planet  are  found  (201). 


VERIFICATION  OF  KEPLER'S  LAWS. 

205.  If  Kepler's  first  two  laws  be  true,  then  the  geocentric 
places  of  the  planets,  computed  by  the  process  that  we  have 
described  ('202),  which  is  founded  upon  them,  ought  to  agree 
with  the  true  geocentric  places  as  obtained  for  the  same  times  by 
direct  observation  ;  .or,  the  heliocentric  places  computed  from  the 
observed  geocentric  places  (177),  ought  to  agree  with  the  same 
as  computed  by  the  elliptic  theory  (200,  201).     Now,  a  great 
number  of  comparisons  have  been  made  between  the  observed 
and  computed  places,  and  in  every  instance  a  close  agreement 
between  the  two  has  been  found  to  subsist.     We  infer,  therefore, 
that  the  motions  of  the  planets  must  be  very  nearly  in  conformity 
with  these  laws. 

The  truth  of  the  third  law  has  been  established  by  a  direct 
comparison  of  the  mean  distances  of  the  different  planets  with 
their  periodic  times. 

Kepler's  laws  have  been  verified  for  the  sun  and  moon,  in  a 
similar  manner. 

206.  The   Relative   Distances   of  the    Snn  or  Moon, 
at  different  times,  result  for  this  purpose,  from  measurements  of 
the  apparent  diameter,  upon  the  principle  that  any  two  distances 
are  inversely  proportional  to  the  corresponding  apparent  diame- 
ters.    Let  A  =  semi-diameter  corresponding  to  the   mean  dis- 
tance, and  $  =  semi-diameter  corresponding  to  any  distance  D : 
then 

S  :  A : :  1 :  D;  whence,  D  =  j. . .  .(46); 

an  equation  which,  when  A  has  been  found,  will  make  known  the 
distance  corresponding  to  any  observed  semi-diameter  <5,  in  terms 
of  the  mean  distance  as  a  unit. 

Now,  to  find  A,  denote  the  greatest  and  least  semi-diameters, 
respectively,  by  5',  5",  and  the  corresponding  distances  by  D'  and 
D"  and  we  have 


and  thence, 

J(D'+D")  = 

whence,  A  =  ^-u=- . . . .  (47). 


130  INEQUALITIES  OF  PLANETARY  MOTIONS. 


CHAPTER  XI. 

INEQUALITIES  OF  THE  MOTIONS  OF  THE  PLANETS  AND  OF  THE 
MOON  ;  TABLES  FOR  FINDING  THE  PLACES  OF  THESE  BODIES. 

2O7.  Gravitation.  It  is  a  general  law  of  nature,  discovered 
by  Sir  Isaac  Newton,  that  bodies  tend  or  gravitate  towards  each 
other,  with  a  force  directly  proportional  to  their  mass,  and 
inversely  proportional  to  the  square  of  their  distance.  The  force 
which  causes  one  body  to  gravitate  towards  another,  is  supposed 
to  arise  from  a  mutual  attraction  existing  between  the  particles 
of  the  two  bodies,  and  is  hence  called  the  Attraction  of  Gravita- 
tion. This  force  of  attraction,  common  to  all  the  bodies  of  the 
Solar  System,  is  the  general  physical  cause  of  their  motions. 
The  sun's  attraction  retains  the  planets  in  their  orbits,  and  the 
planets,  by  their  mutual  attractions,  slightly  alter  each  other's 
motions.  The  reasoning  by  which  Newton's  Theory  of  Universal 
Gravitation  is  established,  appertains  to  Physical  Astronomy, 
and  will  be  presented  in  Part  II. 

2O§.  Perturbations ; — Inequalities.  If  a  planet  were  acted 
on  by  no  other  force  than  the  attraction  of  the  sun,  it  is  proved 
that  its  orbit  would  be  accurately  an  ellipse,  and  the  areas 
described  by  its  radius-vector,  in  equal  times,  would  be  precisely 
equal.  But  it  is  in  reality  attracted  by  the  other  planets,  as  well 
as  the  sun,  and  therefore  its  actual  motions  cannot  be  in  strict 
conformity  with  the  laws  of  Kepler.  In  fact,  if  we  descend  to 
great  accuracy,  the  agreement  between  the  observed  and  com- 
puted places,  noticed  in  Art.  205,  is  found  not  to  be  exact.  The 
deviations  from  the  elliptic  motion,  which  are  produced  by  the 
attractions  of  the  planets,  are  called  Perturbations,  or,  in  Spherical 
Astronomy,  Inequalities.  Although,  as  we  have  just  seen,  the 
fact  of  the  existence  of  inequalities  in  the  motions  of  the  planets 
is  discoverable  from  observation,  their  laws  cannot  be  determined 
without  the  aid  of  theory. 

2O9.  Disturbing  Force.  In  treating  of  the  perturbations 
in  the  motions  of  one  planet,  resulting  from  the  attractions  of 
another,  the  attracting  planet  is  called  the  Disturbing  Body,  and 
the  force  which  produces  the  perturbations  the  Disturbing  Force. 
To  find  the  disturbing  force,  let  P  (Fig.  61)  be  the  planet,  S  the 
sun,  and  M  the  disturbing  body  ;  and  let  PD  represent  the 
attraction  of  M  for  the  planet.  Decompose  PD  into  two  forces, 


COMPONENTS  OF  DISTURBING  FORCE. 


131 


PE  and  PF,  one  of  which,  PE,  is  equal  and  parallel  to  SG,  the 
attraction  of  M  for  the  sun ;  the  other,  PF,  will  be  known  in 
position  and  intensity.  The  two  forces,  PE  and  SG,  being  equal 
and  parallel,  they  cannot  alter  the  relative  motion  of  the  sun 


and  planet,  and  accordingly  may  be  left  out  of  account :  there 
remains,  therefore,  the  component  PF,  which  will  be  wholly 
effective  in  disturbing  this  motion.  This,  then,  is  the  disturbing 
force. 

It  happens  in  the  case  of  each  planet,  that  the  distances  of  some 
of  the  other  planets  are  so  great  that  their  disturbing  forces  are 
insensible.  The  attractions  of  these  bodies  for  the  sun  and  planet, 
when  they  are  exterior  to  the  planet,  are  sensibly  equal  and 
parallel.  Owing  to  the  great  distance  of  the  planets  from  each 
other,  and  the  smallness  of  their  mass  compared  with  that  of  the 
sun,  the  disturbing  force  is  in  every  instance  very  minute  in 
comparison  with  the  sun's  attraction. 

21O.  Components  of  Disturbing  Force ;— their  Effects. 
It  is  plain  that  the  disturbing  force  will,  in  general,  be  obliquely 
inclined  to  the  perpendicular  to  the  plane  of  the  orbit,  PK,  the 
tangent  to  the  orbit,  PT,  and  the  radius- vector,  PS ;  and  may, 
therefore,  be  decomposed  into  forces  acting  along  these  lines. 
The  component  along  the  perpendicular  will  alter  the  latitude, 
and  the  two  others  both  the  longitude  and  radius-vector ;  that 
along  the  tangent  by  changing  the  velocity  of  the  planet,  and 
that  along  the  radius- vector  by  changing  the  gravity  towards  the 
sun.  It  appears,  therefore,  that  the  disturbing  force  produces  at 


132  INEQUALITIES   OF   PLANETARY   MOTIONS. 

the  same  time  perturbations  or  inequalities  of  longitude,  of  latitude, 
and  of  radius-vector. 

211.  Determination  of  Inequalities.     Let  us  now  con- 
sider how  these  inequalities  may  be  determined.     In  the  first 
place,  the  inequalities  produced  by  each  disturbing  body  may 
be  separately  investigated  upon  mechanical  principles,  as  if  the 
other  bodies  did  not  exist;  for  the  reason  that  the  effect  of  each 
disturbing  body  is  sensibly  the  same  that  it  would  be  if  the  other 
bodies  did  not  act.     That  this  is  very  nearly,  if  not  quite  true, 
may  be   at  once   inferred   from  the  minuteness  of  the  whole 
disturbance  produced  by  the  joint  action  of  all  the  disturbing 
forces  of  the  system.     The  problem  which  has  for  its  object  the 
determination  of  the  inequalities  in  the  motions  of  one  body,  in 
its  revolution  around  a  second,  produced  by  the  attraction  of  a 
third,  is  called  the  Problem  of  the  Three  Bodies.     If,  in  the  case 
of  any  one  planet,  this  problem  be  solved  for  each  of  the  other 
bodies  of  the  system  which  occasion  sensible  perturbations,  all 
the  inequalities  to  which  the  motion  of  the  planet  is  subject  will 
become  known. 

The  general  solution  of  the  problem  of  the  three  bodies,  that  is,  for 
any  mass  and  distance  of  the  disturbing  body,  or  any  intensity 
of  the  disturbing  force,  cannot  be  effected  in  the  existing  state 
of  the  mathematical  sciences.  But  the  problem  has  been  solved 
for  the  case  that  presents  itself  in  nature,  in  which  the  disturbing 
force  is  very  minute  in  comparison  with  the  central  attraction. 
The  results  obtained  by  the  analysis  are  certain  analytical  ex- 
pressions for  the  perturbations  in  longitude,  latitude,  and  radius- 
vector,  involving  variables  and  constants. 

212.  Equation!    of  Specific    Inequalities    of    Longi- 
tude.    The  general  expression  for  the  whole  perturbation  in 
longitude,  due  to  the  action  of  any  one  disturbing  body,  is  of  the 
form 

C  sin  A  +  C'  sin  A'  +  C"  sin  A",  etc., 

in  which  C,  C',  C",  etc.,  are  constants,  and  A.  A',  A",  etc.,  angles 
depending  upon  the  positions  of  the  disturbing  and  disturbed 
planets,  with  respect  to  each  other  and  the  sun,  and  also,  in  some 
cases,  with  respect  to  the  nodes  and  perihelia  of  their  orbits. 
Each  of  the  terms,  C  sin  A,  C'  sin  A',  etc.,  is  technically  called 
an  Equation,  and  is  considered  as  representing  a  specific  ine- 
quality. The  variable  angle  whose  sine  enters  into  the  term  is 
called  the  Argument  of  the  inequality,  and  the  constant  is  called 
the  Coefficient  of  the  inequality.  As  the  greatest  value  of  the 
sine  of  the  argument  is  unity,  the  coefficient  is  equal  to  the  great- 
est value  of  the  inequality. 

213.  Calculation  of  Inequalities.     The  value   of  each 
argument  may  be  derived  for  any  assumed  time,  from  the  elliptic 
theory  of  the  planetary  motions ;  and  the  coefficients  of  all  the 


CALCULATION   OF   INEQUALITIES.  133 

inequalities  may  be  calculated  by  making  repeated  determina- 
tions of  the  difference  between  the  observed  and  computed  longi- 
tude of  the  disturbed  planet.  By  putting  the  entire  expression, 
C  sin  A  +  C'  sin  A',  etc.,  equal  to  each  one  of  the  differences  of 
longitude  so  determined,  we  may  form  as  many  equations  as  there 
are  unknown  quantities,  C,  C',  etc.,  from  which  their  values  may 
be  deduced. 

The  coefficient  of  any  inequality  being  known,  the  value  of  the 
inequality,  at  any  particular  time,  will  become  known  if  that  of 
the  argument  be  found.  This  value  will  be  the  correction  for 
that  inequality,  to  be  applied  to  the  elliptic  place  of  the  planet 
computed  for  the  assumed  time. 

214.  Inequalities    of    Latitude    and    Radius-vector. 
The  theory  of  these  inequalities,  and  of  their  computations,  is 
similar  to  that  of  the  inequalities  of  longitude  just  explained. 

215.  Inequalities  are  Periodic.     We  have  seen  that  the 
arguments  of  the  inequalities  are  angles  depending  on  the  con- 
figurations of  the  disturbing  and  disturbed  planets  with  respect 
to  each  other  and  the  sun,  or  with  respect  to  the  nodes  or  peri- 
helia of  their  orbits.     Whenever  these  configurations  become  the 
same,  as  they  will  periodically,  the  arguments,  and  therefore  the 
inequalities  themselves,  will  have  the  same  value.     It  follows, 
therefore,  that  the  inequalities  in  question  are  periodic.  • 

The  interval  of  time  in  which  an  inequality  passes  through  all 
its  gradations  of  positive  and  negative  value,  is  called  the  Period 
of  the  inequality.  It  is  manifestly  equal  to  the  interval  of  time 
employed  by  the  argument  in  increasing  from  zero  to  360° ;  for, 
in  this  interval  sin  A  or  cos  A  takes  all  its  values,  both  positive 
and  negative,  and  at  the  expiration  of  it  recovers  the  same  value 
again. 

216.  Inequalities  of  Elliptic   Element*.      It  has  been 
stated  that  the  elements  of  the  elliptic  orbits  of  the  planets  are, 
for  the  most  part,  subject  to  a  slow  variation  from  century  to 
century.     Investigations  in  Physical  Astronomy  have  established 
that  the  variations  of  the  elements  are  due  to  the  action  of  the 
disturbing  forces  of  the  planets,  and  that  they  are  not  progressive 
(except  in  the  cases  of  the  longitude  of  the  node  ana  the  longi- 
tude of  the  perihelion),  but  are  really  periodic  inequalities  whose 
periods  comprise  many  centuries.     From  the  great  lengths  of 
their  periods  these  inequalities  are  termed  Secular  Inequalities,  in 
order  to  distinguish  them  from  the  inequalities  of  the  elliptic 
motion,  denominated  Periodic  Inequalities,  the  periods  of  which 
are  comparatively  short. 

Physical  Astronomy  furnishes  expressions  called  Secular  Equa- 
tions, which  give  the  value  of  an  element  at  any  assumed  time. 

2 IT.  The  Inequalities  of  the  ?Ioon"s  .fiction  arise  from 
the  disturbing  action  of  the  sun.  The  attractions  of  each  of  the 
planets  for  the  moon  and  earth  are  sensibly  equal  and  parallel. 


134  INEQUALITIES  OF  THE   MOON'S  MOTION. 

The  lunar  inequalities  are  investigated  upon  the  same  principle 
as  the  planetary,  and  are  represented  by  equations  of  the  same 
general  form,  that  is,  consisting  of  a  constant  coefficient  and  the 
sine  or  cosine  of  a  variable  argument.  They  far  exceed  in  num- 
ber and  magnitude  those  of  any  single  planet. 

There  are  three  lunar  inequalities  of  longitude  which  are  promi- 
nent above  the  rest,  and  were  early  discovered  by  observation. 

The  most  considerable  is  called  the  Evection,  and  was  dis- 
covered by  Ptolemy  in  the  first  century  of  the  Christian  era.  It 
has  for  its  argument  double  the  angular  distance  of  the  moon 
from  the  sun  minus  the  mean  anomaly  of  the  moon,  and  amounts 
when  greatest  to  1°  20'  30". 

The  second  is  called  the  Variation,  and  was  discovered  in  the 
sixteenth  century  by  Tycho  Brahe.  Its  argument  is  double  the 
angular  distance  of  the  moon  from  the  sun,  and  its  maximum 
value  is  35'  42". 

The  third  is  denominated  the  Annual  Equation,  from  the  cir- 
cumstance of  its  period  being  an  anomalistic  year.  Its  argument 
is  the  mean  anomaly  of  the  sun. 

The  discovery  of  the  other  lunar  inequalities  (with  the  ex- 
ception of  one  inequality  of  latitude),  is  due  to  Physical  Astro- 
nomy. 

21  §.  Calculation  of  Exact  Heliocentric  Place  of  a 
Planet.  To  present  now  at  one  view  the  entire  process  of  cal- 
culating the  co-ordinates  of  the  exact  heliocentric  place  of  a 
planet,  or  of  the  geocentric  place  of  the  moon,  at  any  assumed 
time, — 

(1).  Seek  the  elements  of  the  elliptic  orbit  from  a  table  of  ele- 
ments, such  as  Table  II.  or  III.,  allowing  for  the  proportional 
part  of  the  secular  variation ;  or  (more  exactly)  obtain  them  from 
their  secular  equations  (216). 

(2).  Compute  the  longitude,  latitude,  and  radius- vector,  by  the 
elliptic  theory  (200,  201). 

(3).  Compute  the  values  of  the  inequalities  in  longitude,  lati-. 
tude,  and  radius- vector,  by  means  of  their  equations  (212,  213, 
214),  and  apply  them  individually,  with  their  proper  signs,  as 
corrections  to  the  elliptic  values  of  the  longitude,  latitude,  and 
radius- vector. 

When  the  exact  heliocentric  place  of  a  planet  has  been  found, 
its  geocentric  place  may  be  determined  by  the  process  referred 
to  in  Art.  202. 

Geocentric  Place  of  the  Sun.  The  elements  of  the  sun's  appa- 
rent orbit  are  the  same  as  those  of  the  earth's  actual  orbit,  except 
that  the  geocentric  longitude  of  the  perigee  of  the  one  exceeds 
the  heliocentric  longitude  of  the  perihelion  of  the  other  by  180°. 
From  these  elements  the  longitude  and  radius- vector  are  obtained 
as  in  Art.  203.  The  values  of  the  inequalities  resulting  from  the 
earth's  motion  are  then  to  be  applied  to  these  as  corrections. 


ASTRONOMICAL  TABLES.  135 


TABLES  OF  THE  SUN,  MOON,  AND  PLANETS. 

219.  The  calculation  of  the  co-ordinates  of  the  place  of  the 
sun,  moon,  or  any  planet,  for  any  assumed  time,  may  be  greatly 
facilitated  by  the  use  of  tables.     The  principle  and  mode  of  con- 
struction of  tables  adapted  to  this  purpose  are  explained  in  Part 
III.     We  will  only  remark  here  that  the  tables  save  the  neces- 
sity of  calculating  the  equations  of  the  inequalities  (218);  since 
they  make  known  their  values  corresponding  to  the  values  of 
the  arguments  at  the  time  supposed.     These  values  of  the  argu- 
ments are  also  readily  obtained  from  tables  especially  designed 
for  this  purpose. 

Tables  of  the  sun,  moon,  and  of  each  of  the  principal  planets, 
have  been  calculated  by  different  astronomers,  and  are  now  in 
general  use. 

220.  Epliemeri§.     With  the  aid  of  these  tables  an  ephemeris 
of  each  body  is  computed,  and  published  for  each  year  in  ad- 
vance, in  the  American  and  English  Nautical  Almanacs.     An 
Ephemeris  of  a  heavenly  body  is  a  collection  of  tables  exhibiting 
the   longitude,   latitude,  right  ascension,  declination,  parallax, 
semi-diameter,  etc.,  of  the  body,  at  stated  periods  of  time,  as  at 
noon  of  each  day  throughout  the  year. 


J36  MOTIONS  OF  THE  COMETS. 


CHAPTER  XII. 

MOTIONS  OF  THE  COMETS. 

221.  Apparent  motions.     "When  first  seen,  a  comet  is  ordi- 
narily at  some  distance  from  the  sun  in  the  heavens,  and  moving 
towards  it.     After  this,  it  continues  to  approach  the  sun,  for 
a  certain  time,  and  then  recedes  to  a  greater  or  less  distance, 
and  finally  disappears.     In  many  instances  comets  have  come 
so  near  the  sun,  as  to  be  for  a  time  lost  in  its  beams.     It  has 
sometimes  happened  that  a  comet  has  not  made  its  appearance 
in  the  firmament  until  after  the  time  of  its  nearest  apparent 
approach  to  the  sun,  and  when  it  is  receding  from  him  in  the 
heavens.     This  was  the  case  with  the  great  comet  of  1843.     It 
was  first  seen,  in  this  country,  in  open  day,  on  the  28th  of  Febru- 
ary, in  the  immediate  vicinity  of  the  sun ;  and  after  this  moved 
away  from  it,  and,  gradually  diminishing  in  brightness,  in  about 
a  month  became  invisible. 

Comets  resemble  the  planets  in  their  changes  of  apparent  place 
among  the  fixed  stars,  but  they  differ  from  them  in  never  having 
been  observed  to  perform  an  entire  circuit  of  the  heavens.  Their 
apparent  motions  are  also  more  irregular  than  those  of  the 
planets,  and  they  are  confined  to  no  particular  region  of  the 
heavens,  but  traverse  indifferently  every  part. 

222.  Orbits  of  Comets.     Sir  Isaac  Newton,  from  observa- 
tions that  had  been  made  upon  the  remarkable  comet  of  1680, 
ascertained  that  this  cornet  described  a  parabolic  orbit,  haying 
the  sun  at  its  focus,  or  an  elliptic  orbit  of  so  great  an  eccentricity 
as  to  be  undistinguishable  from  a  parabola,  and  that  its  radius- 
vector  described  equal  areas  in  equal  times.     Since  then,  the 
orbits  of  240  comets  have  been  computed,  and  found  to  be,  the 
majority  of  them,  of  a  parabolic  form,  or  sensibly  so. 

It  was  demonstrated  by  Newton,  on  the  theory  of  gravitation, 
that  a  body  projected  into  space  may  describe  about  the  sun  as 
a  focus  either  one  of  the  conic  sections,  and  that  the  form  of  the 
orbit  will  depend  upon  the  projectile  velocity  alone.  With  one 
particular  velocity  the  orbit  will  be  a  parabola ;  with  any  less 
velocity  it  will  be  an  ellipse  or  circle;  and  with  any  greater 
velocity  it  will  be  an  hyperbola.  Now,  as  there  is  but  one  velo- 
city from  which  a  parabolic  orbit  will  result,  and  as  any  comet, 
which  may  have,  originally  moved  in  a  hyperbola,  must  have 


COMETS  OF  KNOWN  PERIOD.  137 

passed  its  perihelion,  and  receded  beyond  the  limits  of  the  solar 
system,  it  may  be  inferred,  with  great  probability,  that  the  orbits 
of  the  comets  whose  observed  courses  are  not  distinguishable 
from  parabolic  arcs,  are  in  fact  ellipses  of  great  eccentricity. 
This  is  the  theory  of  the  cometary  motions  proposed  by  Newton. 
The  orbits  of  some  of  the  comets  are  known  from  observation 
to  be  very  eccentric  ellipses. 

223.  Elements  of  Parabolic  Orbit.      The  elements  of 
the  parabolic  path  conceived  to  be  traced  by  a  comet  during  the 
period  in  which  it  remains  visible,  are :  the  longitude  of  the  as- 
cending node,  the  inclination  of  the  orbit,  the  longitude  of  the 
perihelion,  and  the  epoch  of  the  perihelion  passage.     Assuming 
that  the  radius- vector  describes  areas  proportional  to  the  times, 
these  elements  may  be  computed  from  three  observed  geocentric 
places.     But  the  problem  is  one,  of  considerable  difficulty. 

224.  Entire  Elliptic  Orbits.— Periods  of  Revolution. 
Astronomers  do  not  in  general  seek  to  deduce,  from  the  obser- 
vations made  during  one  appearance  of  a  comet,  its  entire  elliptic 
orbit.     It  is  impossible,  from  such  observations,  to  compute  the 
major-axis  of  its  orbit  and  its  period  with  any  accuracy,  inas- 
much as  in  the  interval  during  which  they  are  made,  the  comet 
describes  but  a  small  portion  of  its  entire  orbit.     As  examples 
of  the  uncertainty  of  such  determinations,  four  periods  have  been 
found  by  Bessel  for  the  comet  of  1807,  of  which  the  least  is  1,483 
years  and  the  greatest  1,952  years ;  and  for  the  great  comet  of 
1811  the  two  periods,  2,301  years  and  3,056  years,  have  been 
computed.    The  uncertainty  becomes  much  less  when  the  period 
of  revolution  is  short. 

The  only  mode  of  obtaining  the  period  of  a  comet's  revolution 
with  certainty  is  by  directly  comparing  the  times  of  its  succes- 
sive perihelion  passages.  A  comet  cannot  be  recognized  at  a 
second  appearance  by  its  aspect ;  for  this  is  liable  to  great  altera- 
tions. But  it  may  be  identified  by  means  of  the  elements  of  its 
parabolic  orbit  (223),  as  it  is  extremely  improbable  that  the  ele- 
ments of  the  orbits  of  two  different  comets  will  agree  throughout. 
This  method  of  identifying  a  comet  may  sometimes  fail  of  appli- 
cation, inasmuch  as  the  orbit  of  a  comet  may  experience  great 
alterations  from  the  attractions  of  the  planets. 

225.  Comets   of  Known   Period.      Owing  to  the  great 
lengths  of  the  periods  of  revolution  of  most  of  the  comets,  and 
the  comparatively  short  intervals  of  time  during  which  their 
motions  have  been  carefully  observed,  there  are  but  eight  comets 
whose  periods  and  entire  orbits  have  been  determined  with  cer- 
tainty.    These  have  all  reappeared,  and  in  some  instances  repeat- 
edly, and  verified  the  determinations  of  their  paths  through  space, 
and  the  predictions  of  their  return  to  their  perihelia.     A  comet 
usually  receives  the  name  of  the  astronomer  who  first  determines 
its  orbit  and  period  of  revolution.     The  comets  just  alluded  to 


138 


MOTIONS  OF  THE   COMETS. 


are  designated  as  Halley's,  EnMs,  Bidets,  Faye's,  De  Vico's, 
Brorsorfs,  I? Arrest's,  and  WinnecJce's.  The  last  seven  are  known 
as  Comets  of  Short  Period ;  their  periodic  times  being  comprised 
within  the  limits  of  3.3  years  and  7J-  years.  Their  mean  dis- 
tances from  the  sun  are  less  than  that  of  Jupiter,  and  they  re- 
volve within  the  orbit  of  Saturn.  Halley's  cornet,  in  its  recess 
from  the  sun,  passes  beyond  the  limits  of  the  solar  system,  and 
its  period  approximates  to  that  of  Uranus.  Fig.  62  shows  the 
relative  dimensions  and  positions  of  the  orbits  of  Halley's, 
Encke's,  and  Biela's  comets. 


FIG.  62. 

226.  Comets  whose  Periods  have  been  Approximately 
Calculated.  There  are  a  number  of  cometary  bodies  whose 
periods  of  revolution  and  elliptic  orbits  have  been  approximately 
deduced,  by  calculation,  from  observations  made  at  the  periods 
of  their  first  discovery,  but  which  have  not  since  been  seen. 
Five  of  these  belong  to  the  class  of  comets  of  comparatively  short 
period,  and  small  mean  distance  from  the  su-n  ;  their  computed 
periods  being  from  five  to  seven  years.  Two  have  periods  of  10 
years  and  16  years,  respectively.  Five  form,  with  Halley's 
comet,  a  distinct  class;  their  periodic  times  are  all  about  75 
years,  and  their  mean  distances  from  the  sun  nearly  equal  to  that 
of  Uranus. 

There  are  also  more  than  twenty  comets  whose  entire  elliptic 


NUMBER  OF  COMETS.  139 

orbits  are  believed  to  have  been  ascertained  with  a  certain  degree 
of  approximation  to  the  truth.  Their  mean  distances  exceed  the 
limits  of  the  solar  system,  and  their  periods  are  much  longer  than 
that  of  the  most  distant  planet.  The  same  is  known  to  be  true 
of  the  mean  distances  and  periods  of  all  the  remaining  comets 
that  have  been  carefully  observed. 

227.  All  the  Comets  of  Comparatively  Short  Period 
(viz.,  from  3.3  years  to  16  years)  revolve  around  the  sun  in  the 
same  direction  as  the  planets,  and  like  the  planetoids,  in  planes 
inclined  less  than  35°  to  the  plane  of  the  ecliptic.  But  their 
orbits  are  much  more  eccentric  than  the  orbits  of  the  minor 
planets.  They  form  a  group  of  bodies  whose  orbits  bear  a  strik- 
ing resemblance  to  each  other,  and  occupy  a  position,  in  respect 
to  their  orbital  motions,  intermediate  between  the  planetoids  and 
the  comets  of  long  period  (75  years  and  more).  They  are  com- 
paratively taint  objects,  and  have  generally  been  visible  only 
with  the  aid  of  a  telescope.  All  the  other  comets,  whose  mean 
distance  from  the  sun  does  not  exceed  that  of  the  most  distant 
planet,  with  the  exception  of  Halley's,  also  have  a  direct  motion. 
Some  of  these,  on  their  return  to  their  perihelia,  have  become 
visible  to  the  naked  eye ;  Halley's  comet  conspicuously  so. 

22§.  Comets  of  Long  Period.  Of  220  observed  comets, 
whose  mean  distances  from  the  sun  exceed  that  of  Neptune, 
about  an  equal  number  have  a  direct  and  a  retrograde  motion. 
The  perihelia  of  more  than  two-thirds  of  the  orbits  fall  within 
the  orbit  of  the  earth.  The  aphelia  lie  far  beyond  the  orbit  of 
Neptune.  There  is  little  reason  to  doubt  that  many  comets 
recede  tens  of  thousands  of  millions  of  miles  before  they  begin 
to  return  to  the  sun  again  ;  and  that  the  periods  of  most  of  them 
include  a  number  of  centuries,  and  of  many  of  them  even  tens 
of  centuries.  The  planes  of  their  orbits  are  inclined  under  every 
variety  of  angle  to  the  plane  of  the  ecliptic. 

229.  Comets    of    Small     Perihelion    Distance.      Some 
comets  come  into  close  proximity  to  the  sun.     The  great  comet 
of  1680,  according  to  the  computation   of  Newton,  came  166 
times  nearer  the  sun  than  the  earth  is.     The  no  less  remarkable 
comet  of  1843  approached  still  nearer ;  when  at  its  perihelion,  it 
was  less  than  70,000  miles  from  the  sun's  surface.     Its  orbital 
velocity  at  that  time  was  350  miles  per  second ;  and  it  accom- 
plished a  semi-revolution  around  the  sun  (from  n  to  n',  Fig.  63) 
in  the  astonishingly  short  interval  of  2  hours. 

230.  Number  of  Comets.     The  number  of  recorded  appear- 
ances of  comets  is  about  800,  but  the  actual  number  of  cometary 
bodies  connected  with  the  solar  system  is  undoubtedly  far  greater 
than  this.     This  list  of  recorded  appearances  comprises,  for  the 
great  number  of  years  which  precede  the  date  of  the  inven- 
tion of  the  telescope,  only  those  comets  which  were  very  con- 
spicuous to  the  naked  eye;  giving,  for  example,  only  three  in 


140  MOTIONS  OF  THE   COMETS. 

the  thirteenth,  and  three  in  the  fourteenth  century  ;  and,  since 
the  heavens  have  been  attentively  examined  with  telescopes, 
from  two  to  three  comets,  on  an  average,  have  made  their  appear- 
ance every  year,  of  which  the  great  majority  were  telescopic. 
The  periods  of  these,  as  well  as  of  the  others,  are  in  general  of 
such  vast  length  that  probably  not  more  than  half  the  whole 
number  of  comets  have  returned  twice  to  their  perihelia  during 
the  last  two  thousand  years.  From  these  considerations  it  ap- 
pears, that,  had  the  heavens  been  attentively  surveyed  with  the 
telescope  during  the  last  two  thousand  years,  as  many  as  2,500 
different  cometary  bodies  would  have  been  seen.  But,  as  there 
are  various  causes  which  may  tend  to  prevent  a  comet  from  being 
seen  when  present  in  our  firmament, — as  continued  proximity  to 
the  sun  in  the  heavens,  too  great  distance  from  the  sun  and 
earth,  want  of  intrinsic  lustre,  etc., — it  is  highly  probable  that 
there  are,  in  fact,  many  thousands  of  these  bodies. 


HALLEY'S    COMET. 

231.  Halley's  comet  is  so  called  from  Sir  Edmund  Halley, 
Second  Astronomer  Koyal  of  England,  who  ascertained  its  period, 
and  correctly  predicted  its  return.  From  a  comparison  of  the 
elements  of  the  orbits  described  by  the  cornets  of  1531,  1607, 
and  1682,  he  concluded  that  the  same  comet  had  made  its  appear- 
ance in  these  several  years,  and  predicted  that  it  would  again 
return  to  its  perihelion  towards  the  end  of  1758  or  the  beginning 
of  1759.  Previous  to  its  appearance,  Clairaut,  a  distinguished 
French  astronomer,  undertook  the  arduous  task  of  calculating 
its  perturbations  from  the  disturbing  actions  of  the  planets  during 
this  and  the  preceding  revolution.  He  found,  that,  from  this 
cause,  it  would  be  retarded  about  618  days, — 100  days  from  the 
effect  of  Saturn,  and  518  days  from  the  action  of  Jupiter, — and 
predicted  that  it  would  reach  its  perihelion  within  a  month,  one 
way  or  the  other,  of  the  middle  of  April,  1759.  It  actually 
passed  its  perihelion  on  the  12th  of  March,  1759.  Assuming  the 
earth's  mean  distance  from  the  sun  to  be  unity,  the  perihelion 
distance  of  this  comet  is  0.6,  and  aphelion  distance  35.4.  Accord- 
ingly it  approaches  the  sun  to  within  about  one-half  the  distance 
of  the  earth,  and  recedes  from  him  to  nearly  twice  the  distance 
of  Uranus.  (See  Fig.  62.)  Its  period  is  about  76  years,  but  is 
liable  to  a  variation  of  a  year  or  more  from  the  effect  of  the 
attractions  of  the  planets.  The  inclination  of  its  orbit  is  18°,  and 
its  motion  is  retrograde.  The  last  perihelion  passage  took  place 
on  the  16th  of  November,  1835,  within  a  few  days  of  the  pre- 
dicted time.  The  next  will  occur  in  the  year  1911.  It  is  to- be 
expected  that  the  perturbations  will  now  be  determined  with 


ENCKE'S  COMET.  141 

such  increased  accuracy  that  the  error  in  the  prediction  of  its 
next  perihelion  passage  will  be  less  than  one  day. 

Probable  repeated  appearances  of  this  comet  have  been  traced 
as  far  back  as  the  year  11  B.  C.  It  seems  to  have  been  particu- 
larly conspicuous  in  the  years  1066  and  1456. 


EXCKE'S    COMET. 

t 

232.  This  comet  is  remarkable  for  its  short  period  of  revolu- 
tion, which  is  only  3.3  years.     It  moves  in  an  orbit  inclined 
only  13°  to  the  plane  of  the  ecliptic,  and  whose  perihelion  is  at 
the  distance  from  the  sun  of  the  planet  Mercury,  and  aphelion 
at  a  distance  somewhat  less  than  that  of  Jupiter  (see  Fig.  62). 
Its  period  and  elliptic  orbit  were  determined  on  the  occasion  of 
its  fourth  recorded  appearance,  by  Professor  Encke,  of  Berlin. 
Since  then  it  has  returned  a  number  of  times  to  its  perihelion, 
and  in  every  instance  very  nearly  as  predicted.     At  some  of  its 
returns  it  has  become  visible  to  the  naked  eye.     Its  last  return 
took  place  in  1865 ;  the  next  will  be  in  September,  1868. 

233.  Disturbing  Effects  of  a  Resisting  Medium.     The 
motions  of  this  comet  present  the  anomalous  fact  in  the  solar 
system  of  a  period  continually  diminishing,  and  an  orbit  slowly 
contracting,  from  the  operation  of  some  other  cause  than  the  dis- 
turbing actions  of  the  other  bodies  of  the  system.     Professor 
Encke  found  that  after  allowance  had  been  made  for  all  the  per- 
turbations produced  by  the  planets,  the  actual  time  of  each  peri- 
helion passage  anticipated  the  time  calculated  from  the  duration 
of  the  previous  revolution  about  2|  hours ;  and  that  the  comet 
now  arrives  at  its  perihelion  about  2f  days  sooner  than  it  would 
if  the  period  had  remained  unaltered  since  the  comet  was  first 
seen  in  1786.     This  continual  acceleration  of  the  time  of  the 
perihelion  passage,  discovered  by  Encke,  could  not  be  attributed 
to  the  disturbing  attraction  of  some  unknown  body,  because  this 
attraction  would  produce  other  effects,   which  have  not  been 
noticed.     He  conceived  that  it  could  arise  from  no  other  cause 
than  the  action  of  a  resisting  medium,  or  ether  in  space.     The 
immediate  effect  of  such  a  medium  subsisting  in  the  regions  of 
space  traversed  by  the  comet,  would  be  to  diminish  the  velocity 
in  the  orbit,  which  it  would  at  first  seem  should  delay  the  time 
of  the  perihelion  passage  ;  but  the  velocity  being  diminished,  the 
centrifugal  force  is  weakened,  and  consequently  the  comet  is 
drawn  nearer  to  the  sun,  and  moves  in  an  orbit  lying  within  the 
orbit  due  to  the  sun's  attraction  alone  ;  its  mean  distance  is  there- 
fore diminished,  and  its  period  shortened.     A  similar  pheno- 
menon to  this  is  presented  in  the  oscillations  of  a  pendulum 
freely  suspended.     It  is  well  known  that  the  arc  of  vibration  of 


142  MOTIONS   OF   THE   COMETS. 

the  pendulum  shortens,  and  consequently  its  rapidity  of  oscilla- 
tion increases,  under  the  influence  of  the  resistance  of  the  air. 


BIELA'S  COMET. 

234.  In  February,  1826,  M.  Biela,  of  Josephstadt,  in  Bohemia, 
detected  a  telescopic  comet  in  the  constellation  Aries  ;  and  subse- 
quently made  repeated  observations  upon  its  varying  position  in 
the  heavens.  From  the  results  of  his  observations,  he  calculated 
the  elements  of  its  supposed  parabolic  orbit,  and  found  on  in- 
specting a  catalogue  of  comets  that  the  computed  elements  bore 
a  striking  resemblance  to  those  of  the  comets  of  1772  and  1805. 
He  also  ascertained  that  the  entire  observed  path  of  the  comet 
could  not  be  accurately  represented  by  a  parabolic  orbit,  and 
proceeded  to  compute  from  his  observations  the  elements  of  an 
elliptic  orbit.  He  found  the  period  of  revolution  to  be  6.7  years, 
and  that  it  accorded  with  the  supposition  that  the  same  comet 
had  been  previously  seen  in  1772  .and  1805.  The  period,  as 
since  more  accurately  determined,  is  6.6  years.  Its  orbit  is  in- 
clined 12J°  to  the  plane  of  the  ecliptic;  and  the  perihelion  lies 
just  within  the  orbit  of  the  earth,  while  the  aphelion  falls 
beyond  the  orbit  of  Jupiter  (Fig.  62).  By  a  remarkable  coinci- 
dence, the  orbit  of  this  comet  very  nearly  intersects  the  orbit  of 
the  earth.  At  the  return  of  the  comet  in  1832,  Dr.  Olbers  found 
that  in  going  through  its  descending  node  it  would  pass  within 
20.000  miles  of  the  earth's  orbit,  on  the  inside,  and  that  a  portion 
of  the  orbit  would  fall  within  the  filmy  mass  of  the  comet.  The 
earth  was  more  than  60,000,000  miles  distant  from  the  comet 
at  the  time  of  the  nodal  passage,  and  did  not  reach  the  point  of 
nearest  approach  of  the  two  orbits  until  one  month  after  the 
comet  had  passed  by  it.  In  1805  the  same  comet  passed  within 
6,000,000  miles  of  the  earth. 

According  to  calculation,  the  last  return  of  Biela's  comet  to 
its  perihelion  took  place  in  February,  1866;  but  the  comet  es- 
caped detection.  The  next  return  will  be  in  September,  1872. 


FATE'S  COMET. 

235.  This  comet  was  discovered  and  its  orbit  determined  by 
M.  Faye,  of  the  Paris  Observatory.  Its  period  of  revolution  is 
7^  years.  The  eccentricity  of  its  orbit  (0.556)  is  less  than  that 
of  any  other  known  cometary  body,  although  nearly  twice  as 
great  as  that  of  the  most  eccentric  planetary  orbit. 

The  return  of  this  comet  to  its  perihelion  appears  to  be  accele- 
rated, like  that  of  Encke's  comet,  and  in  a  much  greater  degree, 
by  the  operation  of  a  resisting  medium  in  space.  As  the  perihe- 


LEXELL'S  COMET  or  1770.  148 

lion  distance  of  this  comet  is  much  greater  than  that  of  Encke's, 
it  seems  probable  that  the  resistance  encountered  by  these  comets 
is  due  to  a  collision  with  meteoric  bodies,  or  some  other  form  of 
cosmical  matter. 

The  remaining  comets  of  short  period  need  not  be  specially 
noticed. 


LEXELL'S  COMET  OF  1770. 

236.  It  has  already  been  intimated  that  the  motions  of  the 
comets  are  liable  to  great  derangements,  from  the  operation  of 
the  attractive  forces  of  the  planets.  This  results  from  the  elon- 
gated form  of  the  cornetary  orbits,  in  consequence  of  which  the 
comets,  while  pursuing  their  course  within  the  limits  of  the 
planetary  system,  may  come  into  proximity  to  the  planets,  and 
be  strongly  attracted  by  them.  Halley's  comet  has  already  fur- 
nished an  illustration  of  this  general  fact.  Lexell's  comet  offers 
a  still  more  striking  example  of  the  disturbances  to  which  the 
cometary  motions  are  exposed.  From  observations  made  upon 
this  comet  in  the  year  1770,  Lexell  made  out  that  its  period  was 
5£  years ;  still,  though  a  very  bright  comet,  it  has  not  since  been 
seen.  Burckhardt,  an  eminent  French  calculator,  undertook  to 
investigate  the  cause  of  this  phenomenon,  and  found  that  on  its 
return  to  the  perihelion  in  1776,  the  comet  was  so  situated  with 
regard  to  the  earth  and  sun  as  to  be  continually  hid  by  the  sun's 
rays ;  and  that  in  1779,  before  its  next  return,  it  passed  so  near 
the  planet  Jupiter,  that  his  attraction  was  very  many  times 
greater  than  the  attraction  of  the  sun.  The  consequence  was  that 
its  orbit  was  greatly  enlarged,  so  that  it  no  longer  comes  near 
enough  to  the  earth  to  be  visible. 

Another  fact  to  be  accounted  for  was,  that  the  comet  had  not 
been  seen  previous  to  the  year  1770.  In  seeking  for  its  explana- 
tion it  was  discovered,  by  tracing  back  the  orbit  of  the  comet, 
that  in  1767  it  must  have  passed  near  Jupiter,  and  that  the  action 
of  his  attractive  force  must  have  altered  its  orbit  from  one  of 
large  dimensions  to  the  comparatively  small  orbit,  with  short 
period,  of  the  comet  as  seen  in  1770.  While  describing,  previous 
to  1767,  an  orbit  with  a  large  perihelion  distance,  it  could  not 
have  come  near  enough  to  the  earth  and  sun  to  be  visible. 

This  comet  is  also  remarkable  as  having  made  a  nearer  ap- 
proach to  the  earth  than  any  other  on  record.  On  July  1, 1770, 
its  distance  from  the  earth  was  less  than  1,500,000  miles. 


144 


MOTIONS   OF    THE   COMETS. 


THE  GREAT  COMET  OF  1843. 


237.  This  comet  has  already  been  alluded  to  as  remarkable  for 
having  made  a  nearer  approach  to  the  sun  than  any  other  comet. 
Its  parabolic  path  is  represented  in  Fig.  63.  The  positions  of  the 


FIG.  63. 


comet  at  several  different  dates,  with  the  corresponding  positions 
of  the  earth,  are  also  indicated ;  n  is  the  ascending  and  n'  the 
descending  node.  The  perihelion  is  within  500,000  miles  of  the 
sun's  centre,  and  nearly  midway  between  n  and  n'.  The  incli- 
nation of  the  orbit  is  36°.  the  comet  passed  its  perihelion 
on  February  27,  at  about  5  P.M.  (Philadelphia  time).  On  the 


DONATES  COMET.  145 

28th  it  was  observed  in  full  daylight  in  various  parts  of  New 
England,  in  Mexico,  at  several  places  in  Italy,  and  off  the  Cape 
of  Good  Hope.  It  was  then  about  3°  distant  from  the  sun,  and 
of  a  dazzling  brightness.  Its  great  lustre  at  that  time  doubtless 
resulted  in  part  from  its  tail  being  foreshortened  by  the  obliquity 
under  which  it  was  seen.  After  the  28th  it  showed  itself  with 
great  distinctness  early  in  the  evening,  over  the  western  horizon  ; 
and  though  growing  fainter'  from  night  to  night,  as  it  receded 
from  the  sun,  continued  visible  to  the  naked  eye  until  about  the 
3d  of  April. 

This  comet  is  believed  to  move  in  an  elliptic  orbit  answering 
to  a  period  of  175  years. 


DONATI'S    COMET. 

23§.  This  is  the  great  comet  that  made  its  appearance  in 
1858.  It  was  first  seen  by  Donati  at  Florence,  on  the  2d  of 
June,  1858.  It  was  then  but  a  faint  nebulosity,  discernible  only 
with  a  telescope.  Although  becoming  more  distinct  in  the  field 
of  the  telescope  from  week  to  week,  it  did  not  become  visible 
to  the  naked  eye  until  near  the  1st  of  September.  It  attained 
to  its  greatest  size  and  splendor  after  the  perihelion  passage 
on  September  30,  after  which  it  decreased  in  brightness  as  it 
receded  from  the  sun  and  earth,  moved  off  rapidly  towards 
the  south,  and  finally  disappeared  from  view  in  March,  1859,  in 


FIG.  64 

the  southern  heavens.  Fig.  64  represents  a  portion  of  the  orbit 
of  the  comet,  as  projected  on  the  plane  of  the  earth's  orbit,  and 
several  corresponding  positions  of  the  comet  and  earth.  The 
plane  of  the  orbit  is  inclined  to  that  of  the  earth's  orbit  under  an 
angle  of  63°,  the  portion  of  the  orbit  containing  the  perihelion 

10 


146  MOTIONS  OF  THE  COMETS. 

lying  on  the  north  side  of  the  plane  of  the  earth's  orbit.  When 
first  seen,  on  June  2,  the  comet  was  about  240,000,000  miles 
from  the  earth.  At  the  perihelion  (September  30)  the  distance 
was  less  than  70,000,000  miles.  It  was  at  its  least  distance  from 
the  earth  (nearly  52,000,000  miles)  on  October  10,  but  attained 
its  greatest  brilliancy  five  days  earlier. 

The  period  of  revolution  of  Donati's  comet  has  not  been  deter- 
mined ;  but  it  is  estimated  to  exceed  1,600  years. 


CONSPICUOUS  COMETS  OF  THE  PRESENT  CENTURY. 

239.  These  are,  in  addition  to  Donati's  comet,  and  the  great 
comet  of  1843,  the  great  comet  of  1811,  the  bright  comets  of 
1819,  1825,  and  1835  (Halley's  comet),  and  the  great  comet  of 
1861.  The  comet  of  1811  affords  an  instance  of  a  large  and 
bright  comet,  with  a  perihelion  distance  exceeding  the  earth's 
distance  from  the  sun. 


KEVOLUTION  OF  THE  SATELLITES.  147 


CHAPTER  XIII. 

MOTIONS  OF  THE  SATELLITES. 

240.  As  before  stated,  the  planets  which  have  satellites  are 
Jupiter,  Saturn,  Uranus,  and  Neptune.     The  number  of  Jupiter's 
satellites  is  four,  of  Saturn's  eight,  of  Uranus'  eight,  of  Nep- 
tune's one. 

241.  The  Satellites  of  Jupiter  are  perceptible  with  a  tele- 
scope of  very  low  power.     It  is  found,  by  repeated  observations, 
that  they  are  continually  changing  their  positions  with  respect  to 
one  another  and  the  planet ;  being  sometimes  all  to  the  right  of 
the  planet,  and  sometimes  all  to  the  left  of  it,  but  more  frequently 
some  on  each  side.     They  are  distinguished  from  each  other  by 
the  distance  to  which  they  recede  from  the  planet ;  that  whicn 
recedes  to  the  least  distance  being  called  the  First  Satellite,  that 
which  recedes  to  the  next  greater  distance  the  Second,  and  so  on. 

The  satellites  of  Jupiter  were  discovered  by  Galileo,  in  the 
year  1610. 

The  Satellites  of  Saturn,  Uranus,  and  Neptune  cannot  be  seen, 
except  through  excellent  telescopes.  They  experience  changes 
of  apparent  position,  similar  to  those  of  Jupiter's  satellites. 

242.  The  Satellites  Revolve  around  the  Planet.     The 
apparent  motion  of  Jupiter's  satellites  alternately  from  one  side 
to  the  other  of  the  planet,  leads  to  the  supposition  that  they 
actually  revolve  around  the  planet.     This  inference  is  confirmed 
by  other  phenomena.     While   a   satellite  is  passing  from  the 
eastern  to  the  western  side  of  the  planet,  a  small  dark  spot  is  fre- 
quently seen  crossing  the  disc  of  the  planet  in  the  same  direc- 
tion ;  and  again,  while  the  satellite  is  passing  from  the  western 
to  the  eastern  side,  it  often  disappears,  and,  after  remaining  for  a 
time  invisible,  reappears  at  another  place.     These  phenomena 
are  easily  explained,  if  we  suppose  that  the  planet  and  its  satel- 
lites are  opake  bodies  illuminated  by  the  sun,  and  that  the  satel- 
lites revolve  around  the  planet  from  west  to  east.     On  this  hypo- 
thesis, the  dark  spot  seen  traversing  the  disc  of  the  planet  is  the 
shadow  cast  upon  it  by  the  satellite  on  passing  between  the 
planet  and  the  sun  ;  and  the  disappearance  of  the  satellite  is  an 
eclipse,  occasioned  by  its  entering  the  shadow  of  the  planet. 

As  the  transit  of  the  shadow  occurs  during  the  passage  of  the 
satellite  from  the  eastern  to  the  western  side  of  the  planet,  and 


MOTIONS  OF  THE  SATELLITES. 


the  eclipse  of  the  satellite  daring  its  passage  from  the  western  to 
the  eastern  side,  the  direction  of  the  motion  must  be  from  west 
to  east. 

Analogous  conclusions  may  be  drawn  from  similar  phenomena 
exhibited  by  the  satellites  of  Saturn.  The  satellites  of  Uranus 
also  revolve  around  their  primary ;  but  the  direction  of  their 
motion,  as  referred  to  the  ecliptic,  is  from  east  to  west.  The 
satellite  of  Neptune  revolves  around  the  planet  from  west  to 
east. 

243.  Eclipses.— Transits  of  Shadows.  Let  us  now  exa- 
mine into  the  principal  circumstances  of  the  eclipses  of  Jupiter's 
satellites,  and  of  the  transits  of  their  shadows  across  the  disc  of 
the  primary.  Let  EE'E"  (Fig.  65)  represent  the  orbit  of  the 


Fia.  65. 


earth,  PPT"  the  orbit  of  Jupiter,  and  ss's"  that  of  one  of  its 
satellites,  supposed  to  lie  in  the  plane  of  Jupiter's  orbit.  Sup- 
pose that  E  is  the  position  of  the  earth,  and  P  that  of  the  planet, 
and  conceive  two  lines,  aaf,  bbf,  to  be  drawn  tangent  to  the  sun 
and  planet :  then,  while  the  satellite  is  moving  from  s  to  s'  it  will 
be  eclipsed;  and,  while  it  is  moving  from  /to/'  its  shadow  will 


PERIODS,   MEAN  MOTIONS,   MEAN  DISTANCES.  149 

fall  upon  the  planet.  Again,  if  Ee,  Ee7  represent  two  lines  drawn 
from  the  earth  tangent  to  the  planet  on  either  side,  the  satellite 
will,  while  moving  from  g  to  g',  traverse  the  disc  of  the  planet, 
and,  while  moving  from  h  to  A',  be  behind  the  planet,  and  thus 
concealed  from  view.  It  will  be  seen  on  an  inspection  of  the 
figure,  that,  during  the  motion  of  the  earth  from  E",  the  position  of 
heliocentric  opposition,  to  E7  that  of  conjunction,  the  disappear- 
ances or  immersions  of  the  satellite  will  take  place  on  the  western 
side  of  the  planet ;  and  that  the  emersions,  if  visible  at  all,  can  be 
so  only  when  the  earth  is  so  far  from  opposition  and  conjunction 
that  the  line  Es',  drawn  from  the  earth  to  the  point  of  emersion, 
will  lie  to  the  west  of  Ee.  It  will  also  be  seen,  that,  during  the 
passage  of  the  earth  from  E'  to  E"  the  emersions  will  take  place 
on  the  eastern  side  of  the  planet,  and  that  the  immersions  cannot 
be  visible,  unless  the  line  Fs,  drawn  from  the  earth  to  the  point 
of  immersion,  passes  to  the  east  of  the  planet.  It  appears  from 
observation  that  the  immersion  and  emersion  are  never  both  visi- 
ble at  the  same  period,  except  in  the  case  of  the  third  and  fourth 
satellites. 

If  the  orbits  of  the  satellites  lay  in  the  plane  of  Jupiter's  orbit 
an  eclipse  of  each  satellite  would  occur  every  revolution,  but,  in 
point  of  fact,  they  are  somewhat  inclined  to  this  plane,  from 
which  cause  the  fourth  satellite  sometimes  escapes  an  eclipse. 

244.  Periods.— Mean  .^lotions.— Xeaii  Distances.  The 
periods  and  other  particulars  of  the  motions  of  the  satellites, 
result  from  observations  upon  their  eclipses.  The  middle  point 
of  time  between  the  instants  when  the  satellite  enters  and  enierges 
from  the  shadow  of  the  primary,  is  the  time  when  the  satellite  is  in 
the  direction,  or  nearly  so,  of  a  line  joining  the  centres  of  the  sun 
and  primary.  If  the  latter  continued  stationary,  then  the  inter- 
val between  this  and  the  succeeding  central  eclipse  would  be  the 
periodic  time  of  the  satellite.  But,  the  primary  planet  moving 
in  its  orbit,  the  interval  between  two  successive  eclipses  is  a 
synodic  revolution.  The  synodic  revolution,  however,  being 
observed,  and  the  period  of  the  primary  being  known,  the  peri- 
odic time  of  the  satellite  may  be  computed. 

The  mean  motions  of  the  satellites  differ  but  little  from  their 
true  motions  ;  and  hence  the  forms  of  their  orbits  must  be  nearly 
circular.  The  orbit,  however,  of  the  third  satellite  of  Jupiter  has 
a  small  eccentricity  ;  that  of  the  fourth,  a  larger. 

The  distances  of  the  satellites  from  their  primary,  are  determined 
from  micrometrical  measurements  of  their  apparent  distances  at 
the  times  of  their  greatest  elongations. 

A  comparison  of  the  mean  distances  of  Jupiter's  satellites  with 
their  periodic  times  proves  that  Kepler's  third  law  with  respect 
to  the  planets  applies  also  to  these  bodies  ;  or,  that  the  squares 
of  their  sidereal  revolutions  are  as  the  cubes  of  their  mean  dis- 
tances from  the  primary. 


150  MOTIONS  OF  THE  SATELLITES. 

The  same  law  also  has  place  with  the  satellites  of  Saturn  and 

Uranus. 

245.    The  Computation  of  the  Place  of  a  Satellite 

for  a  given  time,  is  effected  upon  similar  principles  with  that  of 
the  place  of  a  planet.  The  mutual  attractions  of  Jupiter's  satel- 
lites occasion  sensible  perturbations  of  their  motions,  of  which 
account  must  be  taken  when  it  is  desired  to  determine  their 
places  with  accuracy. 

24G.  Relation*  of  Mean  Motion  and  Position.  Laplace 
has  shown  from  the  theory  of  gravitation,  that,  by  reason  of  the 
mutual  attractions  of  the  first  three  of  Jupiter's  satellites,  their 
mean  motions  and  mean  longitudes  are  permanently  connected 
by  the  following  remarkable  relations. 

'  (1.)  The  mean  motion  of  the  first  satellite,  plus  twice  that  of  the 
third,  is  equal  to  three  times  that  of  the  second. 

(2.)  The  mean  longitude  of  the  first  satellite,  plus  twice  that  of  the 
third,  minus  three  times  that  of  the  second,  is  equal  to  180°. 

It  follows,  from  this  last  relation,  that  the  longitudes  of  the 
three  satellites  can  never  be  the  same  at  the  same  time,  and  con- 
sequently that  they  can  never  be  all  eclipsed  at  once. 


INEQUALITY  OF  DATS. 


151 


CHAPTER  XIV. 

THE  SUN,  AND  THE  PHENOMENA  ATTENDING  ITS  APPARENT 

MOTIONS. 

INEQUALITY  OF  DAYS.* 

247.  Suii'§  Motion  relative  to  the  Equator.  We  will 
first  give  a  detailed  description  of  the  sun's  apparent  motion  with, 
respect  to  the  equator,  the  phenomenon  upon  which  the  ine- 
quality of  days  (as  well  as  the  change  of  seasons,  soon  to  be 
treated  of)  immediately  depends. 

Let  YE  AQ  (Fig.  66)  represent  the  equator ;  YTAW  (inclined 
to  YEAQ,  under  the  angle  TOE,  measured  by  the  arc  TE,  equal 
to  23£°),  the  ecliptic ;  TnX  and 
Ww'X',  the  two  tropics;  POP7, 
the  axis  of  the  heavens ;  and 
PEP'Q  the  meridian,  and  HYK  A 
the  horizon,  in  one  of  their  vari- 
ous positions  with  respect  to  the 
other  circles.  About  the  21st 
of  March  the  sun  is  in  the  ver- 
nal equinox  Y,  crossing  the 
equator  in  the  oblique  direction 
YS,  towards  the  north  and  east. 
At  this  time  its  diurnal  circle  is 
identical  with  the  equator ;  and 
it  crosses  the  meridian  at  the  FIG.  66. 

point  E,  south  of  the  zenith  a 

distance  ZE  equal  to  the  latitude  of  the  place.  Advancing 
towards  the  east  and  north,  it  takes  up  the  successive  positions 
S,  S',  S",  etc.,  and  from  day  to  day  crosses  the  meridian  at  r,  r', 
etc.,  farther  and  farther  to  the  north.  Its  diurnal  circles  will  be, 
respectively,  the  northern  parallels  of  declination  passing  through 
S,  S',  S",  etc.,  and  continually  more  and  more  distant  from  the 
equator.  The  distance  of  the  sun,  and  of  its  diurnal  circle  from 
the  equator,  continues  to  increase  until  about  the  21st  of  June, 
when  he  reaches  the  summer  solstice  T.  At  this  point  he  moves 
for  a  short  time  parallel  to  the  equator;  his  declination  changes 
but  slightly  for  several  days,  and  he  crosses  the  meridian  from  day 

*  The  day  here  considered  is  the  interval  between  sunrise  and  sunset. 


152  THE  SUN  AND  ATTENDANT  PHENOMENA. 

to  day  at  nearly  the  same  place.  It  is  on  this  account, — viz., 
because  the  sun  seems  to  stand  still  for  a  time  with  respect  to  the 
equator,  when  at  the  point  90°  distant  from  the  equinox, — that 
this  point  has  received  the  name  of  solstice.*  The  diurnal  circle 
described  by  the  sun  is  now  identical  with  the  tropic  of  Cancer, 
TnX ;  which  circle  is  so  called  because  it  passes  through  T  the 
beginning  of  the  sign  Cancer,  and  when  the  sun  reaches  it  he  is 
at  his  northern  goal,  and  turns  about  and  goes  towards  the  south. f 
The  sun  is,  also,  when  at  the  summer  solstice,  at  its  point  of 
nearest  approach  to  the  zenith  of  every  place  whose  latitude  ZE 
exceeds  the  obliquity  of  the  ecliptic  TE,  equal  to  23^°.  The 
distance  ZT  =  ZE  —  ET  =  latitude  —  obliquity  of  ecliptic.  Dur- 
ing the  three  months  following  the  21st  of  June,  the  sun  moves 
over  the  arc  TA,  crossing  the  meridian  from  day  to  day  at  the 
successive  points  r",  r\  etc.,  farther  and  farther  to  the  south,  and 
arrives  at  the  autumnal  equinox  A  about  the  23d  of  September, 
when  its  diurnal  circle  again  becomes  identical  with  the  equator. 
It  crosses  the  equator  obliquely  towards  the  east  arid  south,  and 
during  the  next  six  months  has  the  same  motion  on  the  south  of 
the  equator,  that  it  has  had  during  the  previous  six  months  on 
the  north  of  the  equator.  It  employs  three  months  in  passing 
over  the  arc  AW,  during  which  period  it  crosses  the  meridian 
each  day  at  a  point  farther  to  the  south  than  on  the  preceding 
day.  At  the  winter  solstice,  which  occurs  about  the  22d  of 
December,  it  is  again  moving  parallel  to  the  equator,  and  its 
diurnal  circle  is  the  same  circle  as  the  tropic  of  Capricorn.  In 
three  months  more  it  passes  over  the  arc  WV,  crossing  the  meri- 
dian at  the  points  6i//,  s',  etc. ;  so  that  on  the  21st  of  March  it  is 
again  at  the  vernal  equinox. 

248.  Explanation  of  Inequality  of  Day*.  The  pheno- 
menon of  the  inequality  of  days  obtains  at  all  places  on  the  earth 
situated  north  or  south  of  the  equator.  At  all  such  places,  the 
observer  is  in  an  oblique  sphere ;  that  is,  the  celestial  equator 
and  the  parallels  of  declination  are  oblique  to  the  horizon.  This 
position  of  the  sphere  is  represented  in  Fig.  11,  p.  22,  where 
HOR  is  the  horizon,  QOE  the  equator,  and  ncr,  set,  etc.,  parallels 
of  declination ;  WOT  is  the  ecliptic.  It  is  also  represented  in 
Fig.  66,  from  which  Fig.  11  differs  chiefly  in  this,  that  the  hori- 
zon, equator,  ecliptic,  and  parallels  of  declination,  which  are 
represented  as  ellipses  in  Fig.  66,  are  in  Fig.  11  projected  into 
right  lines  upon  the  plane  of  the  meridian.  .Since  the  centres  of 
the  parallels  of  declination  are  situated  upon  the  axis  of  the 
heavens,  which  is  inclined  to  the  horizon,  it  is  plain  that  these 
parallels,  as  it  is  represented  in  the  Figs.,  and  as  we  have  before 
seen  (25),  will  be  divided  into  unequal  parts,  and  that  the  dis- 
parity between  the  parts  will  be  greater  in  proportion  as  the 
parallel  is  more  distant  from  the  equator ;  also,  that  to  the  north 
*  From  Sol,  the  sun,  and  sto,  to  stand.  f  From  rpc™,  to  turn. 


INEQUALITY  OF  DAYS.  153 

of  the  equator  the  greater  parts  will  lie  above  the  horizon,  and 
to  the  south  of  the  equator  below  the  horizon.  Now,  the  length 
of  the  day  is  measured  by  the  portion  of  the  parallel  to  the  equa- 
tor, described  by  the  sun,  which  lies  above  the  horizon ;  and  it 
is  evident,  from  what  has  just  been  stated,  that  (as  it  is  shown  by 
the  Fig.)  this  increases  continually  from  the  winter. solstice  W  to 
the  summer  solstice  T,  and  diminishes  continually  from  the  sum- 
mer solstice  T  to  the  winter  solstice  W;  whence  it  appears  that 
the  day  will  increase  in  length  from  the  winter  to  the  summer 
solstice,  and  diminish  in  length  from  the  summer  to  the  winter 
solstice. 

249.  Length  of  Day.     As  the  equator  is  bisected  by  the 
horizon  at  the  equinoxes,  the  day  and  night  must  be  each  twelve 
hours  long.     But,  when   the  sun  is  north  of  the  equator,  the 
greater  part  of  its  diurnal  circle  lies  above  the  horizon,  in  north- 
ern latitudes ;  and  therefore,  from  the  vernal  to  the  autumnal 
equinox,  the  day  is,  in  the  northern  hemisphere,  more  than  12 
hours  in  length.     On  the  other  hand,  when  the  sun  is  south  of 
the  equator,  the  greater  part  of  its  circle  lies  below  the  horizon, 
and  hence  from  the  autumnal  to  the  vernal  equinox  the  day  is 
less  than  12  hours  in  length. 

In  the  latter  interval,  the  nights  will  obviously,  at  correspond- 
ing periods,  be  of  the  same  length  as  the  days  in  the  former. 

250.  Effect§  of  Increa§e  of  Latitude.     The  variation  in 
the  length  of  the  day,  in  the  course  of  the  year,  will  increase 
with  the  latitude  of  the  place ;  for  the  greater  is  the  latitude  the 
more  oblique  are  the  circles  described  by  the  sun  to  the  horizon, 
and  the  greater  is  the  disparity  between  the  parts  into  which  they 
are  divided  by  the  horizon.     This  will  be  obvious,  on  referring 
to  Fig.  11,  p.  22,  where  HOR,  H'OR',  represent  the  positions  of 
the  horizons  of  two  different  places  with  respect  to  these  circles; 
H'OR'  being  the  horizon  for  which  the  latitude,  or  the  altitude 
of  the  pole,  is  the  least. 

For  the  same  reason,  the  days  will  be  the  longer  as  we  proceed 
from  the  equator  northward,  during  the  period  that  the  sun  is 
north  of  the  equinoctial,  and  the  shorter,  during  the  period  that 
he  is  south  of  this  circle. 

251.  Longest  I>ay.     At  ike  equator,  the  horizon  bisects  all 
the  diurnal  circles  (26) ;  and,  consequently,  the  day  and  night  are 
there  each  12  hours  in  length  throughout  the  year. 

At  the  arctic  circle  the  day  will  be  2±  hours  long  at  the  time 
of  the  summer  solstice ;  for  the  polar  distance  of  the  sun  will 
then  be  66 J-°,  which  is  the  same  as  the  latitude  of  the  arctic  cir- 
cle ;  whence  it  follows,  that  the  diurnal  circle  of  the  sun,  at  this 
epoch,  will  correspond  to  the  circle  of  perpetual  apparition  for 
the  parallel  in  question. 

On  the  other  hand,  when  the  sun  is  at  the  winter  solstice,  the 
night  will  be  24  hours  long  on  the  arctic  circle. 


154  THE  SUN  AND  ATTENDANT  PHENOMENA. 

To  the  north  of  the  arctic  circle,  the  sun  will  remain  continually 
above  the  horizon  during  the  period,  before  and  after  the  sum- 
mer solstice,  that  his  north  polar  distance  is  less  than  the  latitude 
of  the  place,  and  continually  below  the  horizon  during  the 
period,  about  the  winter  solstice,  that  his  south  polar  distance  is 
less  than  the  latitude  of  the  place. 

At  the  north  pole,  as  the  horizon  is  coincident  with  the  equator 
(27),  the  sun  will  be  above  the  horizon  while  passing  from  the 
vernal  to  the  autumnal  equinox,  and  below  it  while  passing  from 
the  autumnal  to  the  vernal  equinox.  Accordingly,  at  this 
locality  there  will  be  but  one  day  and  one  night  in  the  course 
of  a  year,  and  each  will  be  of  six  months'  duration. 

252.  In  the  Southern  Hemisphere,  the  circumstances  of 
the  duration  of  light  and  darkness  are  obviously  the  same  as 
in  the  northern,  for  corresponding  latitudes  and  corresponding 
declinations  of  the  sun. 

253.  Problem  I.     The  latitude  of  the  place  and  the  declination 
of  the  sun  being  given,  to  find  the  times  of  the  sun's  rising  and  setting 
and  the  length  of  the  day. 

Let  HPR  (Fig.  67)  be  the  me- 
ridian, HME  the  horizon,  and 
BsD  the  diurnal  circle  described 
by  the  sun.  The  hour  angle  EP*, 
or  its  measure  Etf,  which,  convert- 
ed into  time,  expresses  the  inter- 
val between  the  rising  or  setting 
of  the  sun  and  his  passage  over 
the  meridian,  is  called  the  Semi- 
diurnal Arc.  Now, 

E*  =  EM  +  M*  =  90°  +  M*, 

which  gives 

FIG.  67.  _  .    ,, 

cos  E2  =  —  sin  Mi ; 

and  we  have,  by  Napier's  first  rule, 

sin  M*  =  cot  Ms  tan  ts  =  tan  PMH  tan  EB  =  tan  PH  tan  EB: 

whence,  cos  E*  =  —tan  PH  tan  EB, 

or,     cos  (semi-diurnal  arc)  =  —  tan  lat.  x  tan  dec (48). 

The  semi-diurnal  arc  (in  time)  expresses  the  apparent  time  of 
the  sun's  setting,  and,  subtracted  from  12  hours,  gives  the  appa- 
rent time  of  its  rising.  The  double  of  it  will  be  the  length  of 
the  day. 

In  resolving  this  problem  it  will,  in  practice,  generally  answer 
to  make  use  of  the  declination  of  the  sun  at  noon  of  the  givea 
day,  which  may  be  taken  from  an  ephemeris. 

Exam.  1.  Let  it  be  required  to  find  the  apparent  times  of  the 
sun's  rising  and  setting,  and  the  length  of  the  day  at  New  York, 
at  the  summer  solstice. 


TIME  OF  SUN'S  RISING  OR  SETTING.  155 

Log.  tan  lat.  (40°  42'  40") 9.93474  — 

Log.  tan  dec.  (23°  27'  24") 9.63740 

Log.  cos  (semi-diurnal  arc) 9.57214  — 

Semi-diurnal  arc 111°  55'  26" 

Time  of  sun's  setting 7h.  27m.  42s. 

Time  of  sun's  rising 4    32      18 

Length  of  day 14    55      24 

Exam.  2.  What  are  the  lengths  of  the  longest  and  shortest 
days  at  Boston  ;  the  latitude  of  that  place  being  42°  21'  15"  N  ? 
Ans.  15h.  6m.  25s.,  and  8h.  53m.  35s. 

Exam.  3.  At  what  hours  (apparent  time)  did  the  sun  rise  and 
set  on  May  1,  1866,  at  Charleston  ;  the  latitude  of  Charleston 
being  32°  47',  and  the  declination  of  the  sun  being  15°  9'  30"  1ST? 
Aus.  Time  of  rising,  5h.  19m.  48s. ;  time  of  setting,  6h.  40m. 
12s. 

254.  Problem  II.  To  find  the  time  of  the  surfs  apparent  ris- 
ing or  setting,  the  latitude  of  the  place  and  the  declination  of  the  sun 
being  given. 

At  the  time  of  apparent  rising  or  setting,  the  sun,  as  seen  from 
the  centre  of  the  earth,  will  be  below  the  horizon  a  distance  58 
(Fig.  67)  equal  to  the  refraction  minus  the  parallax.  The  mean 
difference  of  these  quantities  is  34'  45"  (according  to  Bessel).  Let 
it  be  denoted  by  E.  Now,  to  find  the  hour  angle  ZPS  (  =  P), 
the  triangle  ZPS  gives  (see  Appendix), 
7  _  ZP  +  PS  +  ZS  co-lat.  +  co-dec.  +  (90°  +  R)  , 

~2~  ~T~  -••••(49) 

.  ,t_       sin  (k  —  ZP)  sin  (k  —  PS) 

and  sm'iP  = Sr-; — ^    .    '5 '-. 

sm  ZP  sm  PS 

sin  (k — co-lat.)  sin  (k  —  co-dec.) 

5in  -a-i    =: . 7 ; r : ~, n r .  .  .  .  (OU). 

sm  (co-lat.)  sm  (co-dec.) 

The  value  of  P,  in  time,  will  be  the  interval  between  apparent 
noon  and  the  time  of  the  apparent  rising  or  setting  of  the  centre 
of  the  sun's  disc ;  from  which  the  apparent  times  of  the  appa- 
rent rising  and  setting  are  readily  obtained.  To  obtain  the  mean 
times,  these  results  must  be  corrected  for  the  equation  of  time. 

If  the  time  of  the  rising  or  setting  of  the  upper  limb  of  the 
sun,  instead  of  its  centre,  be  required,  we  must  take  for  E  34' 
45"  +  sun's  semi-diameter,  or  50'  47". 

Unless  very  accurate  results  are  desired,  it  will  be  sufficient  to 
take  the  declinations  of  the  sun  at  6  o'clock  in  the  morning  and 
evening.  A  more  accurate  calculation  may  be  made  by  first 
computing  the  times  of  true  rising  and  setting  from  equation 
(48),  and  making  use  of  the  declinations  answering  to  these 
times. 


or,        sin 


156  THE  SUN  AND  ATTENDANT  PHENOMENA. 


TWILIGHT. 

255.  Explanation.  When  the  sun  has  descended  below  the 
horizon,  its  rays  still  continue  to  fall  upon  a  certain  portion  of 
the  body  of  air  that  lies  above  it,  and  are  thence  radiantly  re- 
flected down  to  the  earth,  so  as  to  occasion  a  certain  degree  of 
light;  which  gradually  diminishes  as  the  sun  descends  farther 
below  the  horizon,  and  the  portion  of  air  posited  above  the  hori- 
zon, that  is  directly  illuminated,  becomes  less.  The  same  effect, 
though  in  a  reverse  order,  takes  place  in  the  morning;  previous 
to  the  sun's  rising.  The  light  thus  produced  is  called  the  Cre- 
pusculum  or  Twilight.  The  explanation  of  twilight  will  be  better 
understood  on  examining  Fig.  68,  where  AON  represents  a  por- 


PIG.  68. 

tion  of  the  earth's  surface,  H&B,  the  surface  of  the  atmosphere 
above  it,  and  &mS  a  line  drawn  touching  the  earth  and  passing 
through  the  sun.  The  unshaded  portion,  &cK,  of  the  body  of  air 
which  lies  above  the  plane  of  the  horizon,  HOR,  is  still  illumi- 
nated by  the  sun,  and  shines  down,  by  reflection,  upon  the  station 
of  the  observer  at  0.  As  the  sun  descends,  this  will  decrease, 
until  finally,  when  the  sun  is  in  the  direction  ENS',  it  will  illumi- 
nate directly  none  of  that  part  of  the  atmosphere  which  lies  above 
the  horizon,  and  twilight  will  be  theoretically  at  an  end. 

It  is  assumed  that,  when  the  sun  has  reached  this  position,  in 
which  no  portion  of  air  that  lies  above  the  horizon  is  directly 
illuminated,  faint  stars  will  become  visible  over  the  western 
horizon  ;  and  thus  that  the  end  of  evening  twilight  is  definitely 
marked  by  the  appearance  of  such  stars.  In  like  manner,  morn- 
ing twilight  is  astronomically  defined  as  beginning  when  faint 
stars  situated  in  the  vicinity  of  the  eastern  horizon  begin  to  dis- 
appear. It  has  been  ascertained  from  numerous  observations 
that,  at  the  beginning  of  the  morning  and  end  of  the  evening 
twilight,  as  thus  defined,  the  sun  is  about  18°  below  the 
horizon. 


TWILIGHT.  157 

256.  Approixmate  Determination  of  Height  of  Atmo- 
sphere.    As  we  have  just  seen,  at  the  end  of  evening  twilight, 
the  angle  TRS'  (Fig.  68)  is  equal  to  18°;  H&R  being  the  limit 
of  that  portion  of  the  atmosphere  which  is  capable  of  reflecting 
a  sensible  amount  of  light  to  the  eye,  in  the  direction  RO.     Now, 
if  the  vertical  lines  at  0,  ra,  and  N,  be  produced  to  the  centre  of 
the  earth,  C,  we  shall  have  the  angle  OCN  equal  to  TRS',  or  18°, 
and  therefore  OCR  equal  to  9°.     If,  then,  we  denote  the  radius 
of  the  earth  Cm  by  R,  we  shall  have, 

height  of  atmos.=7ttR=:CR— Cm=Rsec  9°— R^R  (sec  9°  —  1). 
Making  the  calculation,  we  obtain  for  the  height  of  the  atmo- 
sphere, 49.8  miles.  It  is  plain  that  the  actual  height  of  the 
atmosphere  must  be  greater  than  this,  since  a  stratum  of  air  of 
considerable  thickness  may  lie  above  &R,  and  yet  not  have  suffi- 
cient density  to  send  a  sensible  amount  of  reflected  light  to  the 
eye  at  0.  through  the  body  of  air  lying  on  the  line  RO. 

257.  Problem.     The  latitude  of  the  place  and  the  surfs  declina- 
nation  being  given,  to  find  the  time  of  ike  beginning  or  end  of  twi- 
light. 

The  zenith  distance  of  the  sun,  at  the  beginning  of  morning  or 
end  of  evening  twilight,  is  90°  +  I89  ;  we  may  therefore  solve 
this  problem  by  means  of  equations  (49)  and  (50),  taking 
R  =  18°. 

If  the  time  of  the  commencement  of  morning  twilight  be  sub- 
tracted from  the  time  of  sunrise,  the  remainder  will  be  the  dura- 
tion of  twilight. 

258.  Variable  Duration  of  Twilight.     The  duration  of 
twilight  varies  with  the  latitude  of  the  place,  and  with  the  time 
of  the  year.     In  the  northern  hemisphere,  the  summer  are  longer 
than  the  winter  twilights,  and  the  longest  twilights  take  place 
at  the  summer  solstice ;  while  the  shortest  occur  when  the  sun 
has  a  small  southern  declination,  different  for  each  latitude.     The 
summer  twilights  increase  in  length  from  the  equator  northward. 
In  the  southern  hemisphere,  the  phenomena  are  similar  for  cor- 
responding declinations  of  the  sun. 

These  facts  are  consequences  of  the  different  situations  with  respect  to  the  hori- 
zon of  the  centres  of  the  diurnal  circles  described  by  the  sun  in  the  course  of  the 
year,  and  of  the  different  sizes  of  these  circles.  To  make  this  evident,  let  us  con- 
ceive a  circle  to  be  traced  in  the  heavens  parallel  to  the  horizon,  and  at  the  dis- 
tance of  18°  below  it;  this  is  called  the  Cr&pusculum  Circle.  The  duration  of 
twilight  will  depend  upon  the  number  of  degrees  in  the  arc  of  the  diurnal  circle  of 
the  sun,  comprised  between  the  horizon  and  the  crepusculum  circle,  which,  for  the 
sake  of  brevity,  we  will  call  the  arc  of  twilight :  and  this  will  vary  from  the  two 
causes  just  mentioned.  For,  let  Her  (Fig.  69)  represent  the  equator,  and  h'k'r'  a 
diurnal  circle  described  by  the  sun  when  north  of  the  equator;  and  let  hr,  st,  and 
h'r',  s't',  be  the  intersections  of  the  equator  and  diurnal  circle,  respectively,  with 
the  planes  of  the  horizon  and  the  crepusculum  circle.  When  the  sun  is  in  the 
equator,  the  arc  of  twilight  is  hs,  and  when  he  is  on  the  parallel  of  declination 
h'k'r'  it  is  tis'.  Draw  the  chords  hs,  h's',  mn,  and  the  radii,  es,  cs',  or',  en,  cp.  The 
angle  r'h's1  is  the  half  of  /cs',  and  the  angle  pmn  is  the  half  of  pen ;  but  r'cs'  is  less 
than  pen,  and  therefore  r'h's'  is  less  than  pmn.  Again,  chs  is  the  half  of  res,  and 


158 


THE  SUN  AND  ATTENDANT  PHENOMENA. 


FIG.  69. 


therefore  greater  than  pmn,  the  half  of  the  less  angle  pen.  Whence  it  appears 
that  the  chord  h's'  is  more  oblique  to  the  horizon,  and  therefore  greater  than  the 
chord  inn,  and  this  more  oblique  and  greater  than  the  chord  hs.  It  follows,  there- 
fore, that  the  arc  h's'  is  greater,  and 
contains  a  greater  number  of  degrees 
than  the  arc  mn,  and  that  this  arc  is 
greater,  than  ?is.  Thus,  as  the  sun  re- 
cede? from  the  equator  towards  the 
nort\  the  arc  of  twilight,  and  there- 
fore the  duration  of  twilight,  increases 
from  two  causes,  viz. :  1  st.  The  increase 
in  the  distance  of  the  line  of  intersec- 
tion of  the  horizon  with  the  diurnal 
circle  from  the  centre  of  the  circle ; 
and,  2d.  The  diminution  in  the  size  of 
the  circle.  The  change  will  manifestly 
be  greater  in  proportion  as  the  latitude 
is  greater. 

"When  the  sun  is  south  of  the  equa- 
tor, twilight  will,  for  the  same  declina- 
tion, be  shorter  than  when  he  is  north 
of  the  equator,  because,  although  the 
diurnal  circle  will  be  of  the  same  size, 
and  its  intersection  with  the  horizon  at  the  same  distance  from  its  centre,  on  the 
opposite  side,  the  intersection  with  the  crepusculum  circle  will  now  fall  between 
the  intersection  with  the  horizon  and  the  centre,  and  therefore,  by  what  has  just 
been  demonstrated,  the  arc  of  twilight  will  be  shorter. 

The  shortest  twilight  occurs  when  the  sun  is  somewhat  to  the  south  of  the 
equator,  because  the  arc  of  twilight,  for  a  time,  decreases  by  reason  of  the  diminu- 
tion of  its  obliquity  to  the  horizon  more  than  it  increases  in  consequence  of  the 
decrease  in  the  size  of  the  diurnal  circle.  That  the  obliquity  of  the  arc  of  twilight, 
or  rather  of  the  chord  of  the  arc,  to  the  horizon  diminishes,  for  a  time,  when  the 
snn  gets  to  the  south  of  the  equator,  will  appear  from  this,  viz.,  that  the  chord  is 
perpendicular  to  the  horizon  when  the  centre  of  the  diurnal  circle  is  midway  be- 
tween the  horizon  and  the  crepusculum  circle ;  which  will  happen  when  the  sun 
is  a  certain  distance  south  of  the  equator,  varying  with  the  inclination  of  the  axis 
of  the  heavens  to  the  plane  of  the  horizon,  and  therefore  with  the  latitude  of  the 
place. 

The  difference  in  the  length  of  the  summer  and  winter  twilights,  resulting  from 
the  causes  above  specified,  is  augmented  by  the  inequality  in  the  height  of  the 
atmosphere.  Twilight  also  increases  in  length  with  the  obliquity  of  the  sphere. 

259.  Twilight  in  Low  and  Middle  Latitude*.     At  the 

equator,  the  shortest  astronomical  twilight  occurs  at  the  equi- 
noxes, and  is  Ih.  12m.  in  duration.  At  latitude  41°,  it  occurs 
when  the  sun  is  about  6°  south  of  the  equator,  and  continues 
about  1J  hours.  At  the  polar  circle  it  happens  when  the  sun  is 
84°  south  of  the  equator,  and  continues  over  3  hours.  The 
longest  twilight  at  the  equator  is  Ih.  19m. ;  and  at  latitude  41°, 
is  2h.  3m.  in  duration. 

260.  Twilight  in  High  Latitudes.     At  the  latitude  49°, 
the  sun  at  the  time  of  the  summer  solstice  is  only  18°  below  the 
horizon  at  midnight ;  for  the  altitude  of  the  pole,  on  the  parallel 
of  49°,  differs  only  18°  from  the  polar  distance  of  the  sun,  at  this 
epoch.     This  may  be  illustrated  by  Fig.  66,  p.  151,  taking  X  as 
the  point  of  passage  of  the  sun  across  the  inferior  meridian,  and 
supposing  PH  to  be  equal  to  49°.     At   the  summer  solstice, 
PX  =  67° ;  and  thus  the  distance  of  the  sun  below  the  horizon 


THE  SEASONS.  159 

at  midnight  =  HX  =  PX  —  PH  =  67°  —  49°  =  18°.  At  this 
latitude,  therefore,  evening  and  morning  twilight  will  each  con- 
tinue half  the  night,  at  the  summer  solstice,  and  therefore  nearly 
4  hours. 

At  higher  latitudes  than  49°,  twilight  (evening  and  morning) 
will  continue  all  night  for  a  certain  period  of  time  before  and 
after  the  summer  solstice,  during  which  the  polar  distance  of  the 
sun  is  less  than  the  latitude  augmented  by  18°.  At  the  polar 
circle,  this  will  be  the  case  for  2£  months  before  and  2J-  months 
after  the  summer  solstice. 

To  the  north  of  the  arctic  circle,  as  far  as  84°  of  latitude,  during 
the  long  night  that  prevails  before  and  after  the  winter  solstice, 
there  should  be  more  or  less  of  a  twilight  over  the  southern  hori- 
zon, about  the  hour  of  noon  of  every  day  of  24  hours. 

At  either  pole  twilight  commences  about  a  month  and  a  half  before 
the  sun  appears  above  the  horizon,  and  lasts  about  a  month  and  a 
half  after  he  has  disappeared.  For,  since  the  horizon  at  the  pole  is 
identical  with  the  celestial  equator,  the  twilight  which  precedes 
the  long  day  of  six  months  will  begin  when  the  sun  in  approach- 
ing the  equator,  upon  the  other  side,  attains  to  a  declination  of 
18° ;  and  this  will  be  about  50  days  before  he  reaches  the  equa- 
tor, and  rises  at  the  pole.  The  evening  twilight  will  continue,  in 
like  manner,  until  the  sun  has  descended  18°  below  the  equator. 

It  should  be  observed,  with  reference  to  the  above  results, 
that  the  assumed  limiting  angle  of  depression  of  the  sun  below 
the  horizon,  18°,  having  been  determined  from  observations  in 
the  middle  latitudes,  is  probably  too  great  for  high  latitudes ; 
and  also  that  the  astronomical  twilight  above  considered,  is 
much  longer  than  what  is  ordinarily  regarded  as  the  period  of 
twilight. 

THE  SEASONS. 

5261.  General  Explanation  of  Change  of  Sea§ons.  The 

amount  of  heat  received,  at  any  place  on  the  earth,  directly  from 
the  sun,  in  the  course  of  24  hours,  depends  upon  two  operative 
causes  ;  the  length  of  time  that  the  sun  remains  above  the  hori- 
zon, and  the  obliquity  of  its  rays  at  noon.  By  reason  of  the 
obliquity  of  the  ecliptic,  both  of  these  general  circumstances  vary 
materially  in  the  course  of  the  year;  whence  arises  a  variation 
of  temperature,  or  a  change  of  seasons.  Since  the  obliquity  of 
the  ecliptic  is  a  consequence  of  the  inclination  of  the  earth's  axis 
to  the  perpendicular  to  the  plane  of  the  orbit,  the  inclined  posi- 
tion of  the  axis  is  the  primary  cause  of  the  change  of  seasons. 

262.  Climatic  Zones.  The  tropics  and  polar  circles  divide 
the  earth  into  five  parts,  called  Zones,  throughout  each  of  which 
the  yearly  change  of  temperature  is  occasioned  by  a  similar 
change  in  the  circumstances  of  the  sun's  thermal  action.  The 


160  THE  SUN   AND  ATTENDANT  PHENOMENA. 

part  contained  between  the  two  tropics  is  called  the  Torrid  Zone  ; 
the  two  parts  between  the  tropics  and  polar  circles  are  called  the 
Temperate  Zones  ;  and  the  other  two  parts,  within  the  polar  cir- 
cles, are  called  Frigid  Zones. 

At  all  places  in  the  north  temperate  zone,  the  sun  will  always 
pass  the  meridian  to  the  south  of  the  zenith  ;  for  the  latitudes 
of  such  places  exceed  23£°,  the  greatest  declination  of  the  sun 
(see  Fig.  6ft).  The  meridian  zenith  distance  will  be  greatest  at 
the  winter  solstice,  when  the  sun  has  its  greatest  southern  decli- 
nation ;  and  it  will  vary  continually  between  the  values  which 
obtain  at  the  solstices.  The  day  will  be  longest  at  the  summer 
solstice,  and  shortest  at  the  winter  solstice,  and  will  vary  in 
length  progressively  from"  the  one  date  to  the  other. 

We  infer,  therefore,  that  throughout  the  zone  in  question  the 
greatest  amount  of  heat  will  be  received  from  the  sun  at  the 
summer  solstice,  and  the  least  at  the  winter  solstice  ;  and  that 
the  amount  received  will  gradually  increase,  or  decrease,  from 
one  of  these  epochs  to  the  other.  The  solstices  are  not,  however, 
the  epochs  of  maximum  and  minimum  temperature,  but  are 
found. from  observation  to  precede  these  by  about  a  month.  The 
reason  of  this  circumstance  is,  that  the  earth  continues  for  a 
month,  or  thereabouts,  after  the  summer  solstice  to  receive  dur- 
ing the  day  more  heat  than  it  loses  during  the  night,  and  for 
about  the  same  length  of  time  after  the  winter  solstice  continues  to 
lose  during  the  night  more  heat  than  it  receives  during  the  day. 

Within  the  torrid  zone,  the  length  of  the  day  varies  after  the 
same  manner  as  in  the  temperate  zone,  though  in  a  less  degree  ; 
but  the  motion  of  the  sun  with  respect  to  the  zenith  is  different. 
At  all  places  in  the  torrid  zone  the  sun  passes  the  meridian 
during  a  certain  portion  of  the  year  to  the  south  of  the  zenith, 
and  during  the  remaining  portion  to  the.  north  of  it ;  for  all 
places  so  situated  have  their  zeniths  between  the  tropics  in  the 
neavens,  and  the  sun  moves  from  one  tropic  to  the  other,  and 
back  again  to  its  original  position,  in  a  tropical  year.  Through- 
out the  torrid  zone,  therefore,  the  sun  will  be  in  the  zenith  twice 
in  the  course  of  the  year,  and  will  be  at  its  maximum  distance 
from  it  on  the  one  side  and  the  other  at  the  solstices. 

An  inhabitant  of  the  equator,  or  its  vicinity,  will  have  summer 
at  the  two  periods  when  the  sun  is  in  the  zenith,  and  winter  (or 
a  period  of  minimum  temperature)  both  at  the  summer  and  win- 
ter solstice.  Near  the  tropic,  there  will  be  but  little  variation  in 
the  daily  amount  of  heat  received,  during  the  period  that  the 
sun  is  north  of  the  zenith. 

At  the  frigid  zone,  a  new  cause  of  a  change  of  temperature 
exists;  the  sun  remains  continually  above  the  horizon  for  a 
greater  or  less  number  of  days  about  the  summer  solstice,  and 
continually  below  it  for  the  same  number  of  days  about  the  win- 
ter solstice. 


THE  SEASONS. 


161 


263.  General  Effects  of   Increase  of    Latitude.     The 

amount  of  the  yearly  variation  of  temperature  increases  with  the 
latitude  of  the  place;  for  the  greater  is  the  latitude  the  greater 
will  be  the  variation  in  the  length  of  the  day.  Also,  the  mean 
yearly  temperature  is  lower  as  we  recede  from  the  equator  and  ap- 
proach the  poles ;  for  since  the  sun  is,  in  the  course  of  the  year, 
the  same  length  of  time  above  the  horizon,  at  all  places,  the  mean 
yearly  temperature  must  depend  altogether  upon  the  mean  obli- 
quity of  the  sun's  rays  at  noon,  and  this  increases  with  the  latitude. 

264.  Special  Causes  of  Change  of  Cliaiiate.     It  is  im- 
portant to  observe,  that  although,  in  the  main,  climate  varies 
with  the  latitude,  after  the   manner  explained  in  the  foregoing 
articles,  it  is  still  dependent,  more  or  less,  upon  local  circum- 
stances, such  as  the  vicinity  of  lakes,  seas,  or  mountains,  prevail- 
ing winds  of  some  particular  direction,  etc.     As  the  result  of  the 
operation  of  such  special  causes  two  places  may  be  situated  on 
the  same  parallel  of  latitude,  and  yet  have  climates  quite  different. 
Such  differences  of  thermal  condition  are  very  marked  on  the  oppo- 
site coasts  of  the  Atlantic  Ocean,  in  the  middle  and  high  latitudes. 

265.  Seasons  Astronomically  Defined  : — Comparative 
Lengths.     In  the  north  temperate  zone,  Spring,  Summer,  Au- 
tumn, and  Winter,  the  four  seasons  into  which  the  year  is  divided, 
are  considered  as  respectively  commencing  at  the  times  of  the 
Vernal  Equinox,  Summer  Solstice,  Autumnal  Equinox,  and  Winter 
Solstice. 

Let  Y  (Fig.  70)  represent  the  vernal,  and  A  the  autumnal 


S   the 


FIG.  70. 


and  W  the  winter   solstice.     The 


equinox;  »  the  summer, 
perigee  of  the  sun's  apparent  orbit  is  at  present  10°  40'  to  the 
east^of  the  winter  solstice.  Let  P  denote  its  position.  The 
lengths  of  the  seasons  are,  agreeably  to  Kepler's  law  of  areas, 

11 


162  THE  SUN  AND  ATTENDANT  PHENOMENA. 

respectively  proportional  to  the  areas  YES,  SEA,  AEW,  and 
WEV.  Thus,  the  winter  is  the  shortest  season,  and  the  summer 
the  longest;  and  spring  is  longer  than  autumn.  Spring  and 
summer,  taken  together,  are  about  eight  days  longer  than  autumn 
and  winter  united. 

266.  Secular  Variation  of  Length  of  Seasons.    Since  the 
perigee  of  the  sun's  orbit  has  a  progressive  motion,  the  relative  lengths  of  the 
seasons  above  defined  must  be  subject  to  a  continual  variation.     At  the  beginning 
of  the  year  18oO.  the  longitude  of  the  sun's  perigee  was  279°  29'  56".     If  from 
this  we  take  180°,  the  longitude  of  the  autumnal  equinox,  the  remainder,  99°  29' 
56",  is  the  distance  of  the  perigee  from  the  autumnal  equinox  at  that  epoch.     The 
motion  of  the  perigee  in  longitude  is,  at  the  present  date,  at  the  rate  of  61  ".70  per 
year.     Dividing  99°  29'  56"  by  61  ".70,  the  quotient  is  5,805  years.     Hence  it  ap- 
pears that  if  the  annual  motion  of  the  perigee  had  been  constantly  equal  to  61". 7, 
about  5,800  years  anterior  to  1800  the  perigee  would  have  coincided  with  the  au- 
tumnal equinox.     But  the  motion  of  the  perigee  has  in  fact  been  very  different  in 
different  centuries  ;  and  it  appears  from  the  calculations  of  Leverrier  that  10,000 
years  before  the  beginning  of  the  present  century,  the  perigee  was  still  78°  to  the 
east  of  the  autumnal  equinox,  and  that  the  two  points  were  in  approximate  coinci- 
dence 20,000  years  earlier. 

267.  Secular  Vari;ition§  of  Temperature.    The  eccentricity 
of  the  earth's  orbit  is  so  small  (0.017)  that  the  present  annual  change  in  the  sun's 
distance  from  the  earth  has  but  little  effect  in  producing  a  variation  of  tempera- 
ture upon  the  earth's  surface.     The  annual  change  of  its  heating  power  from  this 
cause  amounts  to  no  more  than  one-fifteenth.     So  far  as  this  cause  operates,  it 
makes  the  winters  warmer  and  the  summers  colder  in  the  northern  hemisphere. 
But  the  eccentricity  has  not  always  had  its  present  small  value ;  it  has  been  for 
ages  slowly  diminishing  from  a  certain  maximum  value.     Recent  calculations  made 
by  Leverrier  and  other  eminent  computers,  have  made  known  its  value  at  inter- 
vals of  10,000  years,  or  50,000  years,  for  a  period  of  1,000,000  years  previous  to 
the  beginning  of  the  present  century.     From  these  results,  it  appears  that  it  has 
increased  and  decreased  during  alternate  periods  comprising,  in  general,  about 
50,000  years;  and  that  its  recurring  maximum  value  has  fluctuated  generally  be- 
tween the  limits  .05  and  .075,  while  its  minimum  value  has  been  about  .01.     The 
highest  maximum  value  occurred  850,000  years  since.     The  most  recent  maximum 
occurred  20,000  years  ago,  and  was  only  .019.     At  the  epoch  of  the  highest 
maximum,  the  earth  reached  its  perihelion  during  the  summer  (civil  reckoning) 
in  the  northern  hemisphere,  and  its  aphelion  during  the  winter.     At  that  epoch 
the  heating  power  of  the  sun  was,  by  reason  of  the  eccentricity  of  the  earth's  orbit, 
about  one-fourth  greater  at  the  beginning  of  summer  than  at  the  beginning  of 
winter ;    and  the  midwinter  temperature,  owing  to  the  greater  distance  of  the 
Bun,  was  much  lower  than  at  present.     In  an  article  in  the  Philosophical  Maga- 
zine for  February,  1867,  by  James  Croll,  it  is   computed  that  the  midwinter 
temperature  of  Scotland  was  not  less  than  45°  F.  lower  than  at  present;  and  that, 
at  the  same   epoch,  the  midsummer  temperature  was   correspondingly  higher. 
It  is  also  maintained  that,  by  a  diversion  of  the  gulf-stream,  the  midwinter  tem- 
perature may  have  been  reduced  many  degrees  lower ;  and  that  this  incidental 
effect  of  the  great  eccentricity  of  the  earth's  orbit,  at  that  remote  period,  may 
have  been  the  determining  cause  of  the  glacial  epoch  of  the  earth's  geological  history. 

FORM  AND  DIMENSIONS  OF  THE  SUN. 

268.  The  sun  presents  the  appearance  of  a  luminous  circular 
disc ;  but  it  does  not  follow  from  this  that  its  surface  must  be 
really  flat,  for  such  is  the  appearance  of  all  globular  bodies  when 
viewed  at  a  great  distance.  It  is  ascertained  from  observations 
with  the  telescope  that  the  sun  has  a  rotatory  motion  ;  this  being 
the  fact,  its  surface  must  in  reality  be  of  a  spherical  form ;  for 


DIMENSIONS  OF  SUN.  163 

otherwise  it  would  not,  in  presenting  all  its  sides,  always  appear 
under  the  form  of  a  circle. 

No  sensible  difference  between  the  equatorial  and  polar  diame- 
ters of  the  sun  can  be  detected  by  the  nicest  micrometrical  mea- 
surements. 

269.  Dimensions   of   Sun.     The   sun's   real   diameter   is 
calculated  from  his  apparent  diameter  and  horizontal  parallax. 
Let  ACB  (Fig.  71)  represent 

the   sun,    or    other    heavenly 

body,  and  E  the  place  of  the 

earth  ;  and  let  §  =  AEB,  the 

sun's  apparent  diameter ;  d  = 

2  AS,  its  real  diameter ;  D  = 

ES,  its  distance  from  the  earth ;  FIG.  71. 

and   E,  =  the  radius   of    the 

earth.     We  have,  from  the  triangle  AES, 

AS  =  ES  sin  JAEB,  or  2AS  =  2ES  sin  |AEB  ; 
and  thus  d  =  2D  sin  £$ : 

but  (equa.  7),  D  =_*_', 

whence,       d  =  2R  4^  =  2R  g  =  2R  A  (nearly) ....  (51). 

The  apparent  diameter  of  the  sun  at  the  mean  distance  is  32' 
0",  and  the  corresponding  equatorial  horizontal  parallax  is  8".95. 
Accordingly  we  have,  for  the  real  diameter  of  the  sun  (by  equa. 
51,  whether  the  sines  be  taken  or  the  arcs) 

d  =  2R  x  107.263  =  7925m.60  x  107.263  =  850,123  miles. 

The  mean  diameter  of  the  earth  is  7,912.40  miles  ;  the  diame- 
ter of  the  sun  exceeds  this  in  the  ratio  of  107.442  to  1.  The 
volume  of  the  sun  then  exceeds  that  of  the  earth  in  the  propor- 
tion of  (107.442)3  to  I8,  or  1,240,285  to  1.  The  surface  of  the 
sun  bears  to  that  of  the  earth  the  ratio  of  (107.442)3  to  1s,  or 
11,544  to  1. 

If  models  were  constructed  to  show  the  comparative  dimen- 
sions of  the  sun  and  earth,  and  the  earth  were  represented  by  a 
ball  one  inch  in  diameter,  the  sun  would  be  represented  by  a 
globe  nine  feet  in  diameter.  Perhaps  a  juster  conception  of  the 
enormous  bulk  of  the  sun  may  be  obtained  from  the  considera- 
tion, that,  if  the  centre  of  the  sun  were  coincident  with  the  centre 
of  the  earth,  its  mass  would  extend  nearly  200,000  miles  beyond 
the  orbit  of  the  moon. 

270.  General  Principle.     From  equation  (51)  we  may  de- 
five  the  proportion 

d  :  2E  : :  $  :  2H. 

Thus,  the  real  diameter  of  a  heavenly  body  is  to  the  diameter  of  the 


THE   SUN   AND   ATTENDANT  PHENOMENA. 

earth,  as  the  apparent  diameter  of  the  body  is  to  double  its  horizontal 
parallax. 

SUN'S  SPOTS,  AND  KOTATION  ON  ITS  AXIS.— PHYSICAL 
CONSTITUTION  OF  THE  SUN. 

271.  "When  the  sun  is  viewed  with  a  good  telescope,  provided 
with  colored  glasses  to  protect  the  eye,  black  spots,  or  maculce, 
of  an  irregular  form,  surrounded  by  a  dark  border  of  a  nearly 
uniform  shade,  called  a  penumbra,  are  often  seen  on  its  disc  (Fig. 
72).  Sometimes  several  spots  are  included  within  the  same 


FIG.  72. 


penumbra.  On  the  other  hand,  a  large  penumbra  has  occasion- 
ally been  seen  without  any  central  black  spot.  The  spots  usually 
appear  in  clusters,  composed  of  various  numbers,  from  two  to 
sixty  or  seventy.  It  is  even  said,  that  as  many  as  200  have 
been  counted,  in  one  instance,  in  a  single  group. 

In  most  of  the  individual  spots,  the  central  spot,  or  umbra,  is 
not  perfectly  black ;  but  a  black  nucleus  is  observed  in  the  ma- 
jority of  the  large  and  symmetrical  spots,  to  occupy  some  part 
of  the  umbra,  generally  the  centre.  This  distinction  of  shade  in 
the  umbra  has  been  overlooked  by  most  observers. 

The  penumbra  has  almost  always  a  perceptibly  darker  shade 
at  its  outer  edge  than  in  any  other  part ;  and  its  light  generally 
increases  somewhat  to  its  inner  edge. 

272.  Magnitude  of  the  Spots.  The  absolute  magnitude 
of  the  solar  spots  is  often  very  great.  Spots  are  not  unfrequently 
seen  that  subtend  an  angle  of  I/,  or  60".  Now  the  apparent 
diameter  of  the  earth,  as  viewed  at  the  distance  of  the  sun,  is 
equal  to  double  the  sun's  horizontal  parallax,  or  18"  ;  the  breadth 
of  such  spots  must  therefore  exceed  three  times  the  diameter  of 
the  earth,  or  24,000  miles.  Spots  have  been  observed  whose 


SUN  S  SPOTS.  105 

linear  diameter  was  more  than  45,000  miles,  and  which  were 
therefore,  in  area,  eight  times  as  large  as  the  entire  surface  of  the 
earth.  Some  spots  have  attained  to  even  a  greater  size  than  this, 
and  become  visible  to  the  naked  eye.  A  spot  was  seen  in  June, 
1843,  that  continued  visible  to  the  naked  eye  for  a  whole  week, 
and  which,  according  to  the  measurements  of  M.  Schwabe,  of 
Dessau,  had  a  breadth  of  74,000  miles.  A  group  of  spots,  with 
the  penumbra  surrounding  it,  will  frequently  cover  a  still  larger 
portion  of  the  sun's  disc.  One  noticed  in  April,  1845,  had  a 
linear  extent  of  147,000  miles. 

273.  Variability  of  the  Spot*.     The  form  and  size  of  the 
spots  are  subject  to  rapid  and  almost  incessant  variations.    When 
watched  from  day  to  day,  or  even  from  hour  to  hour,  they  are 
seen  to  enlarge  or  contract,  and  at  the  same  time  to  change  their 
form.     They  sometimes  vanish  in  an  incredibly  short  space  of 
time,  while  others  make  their  appearance  as  suddenly.     Some 
spots  disappear  almost  immediately  after  they  become  visible; 
others  remain  for  weeks  or  even  months.     When  a  spot  disap- 
pears, it  usually  contracts  into  a  point,  and  vanishes  before  the 
penumbra,  which  'gradually  closes  in  upon  it.     When  a  new  spot 
is  developed,  it  is  not  till  it  has  attained  some  measurable  size  that 
a  penumbra  begins  to  be  perceived  distinct  from  the  umbra.   The 
black  nucleus  within  the  umbra  makes  its  appearance  still  later. 
The  spot  usually  grows  very  rapidly,  and  often  attains  its  full  size 
in  less  than  a  day.     During  the  period  of  increase,  and  while  it 
remains  without  material  change  of  size,  its  edges  are  sharply  de- 
fined, and  the  penumbra  exhibits  a  general  uniformity  of  shade. 
Several  observers  have,  however,  distinctly  noticed  a  radiated 
appearance  in  the  penumbral  fringes,  as  if  they  were  traversed 
by  bright  veins  diverging  from  the  central  spot.     In  the  act  of 
decreasing,  the  edges  of  the  spot  are  less  strongly  defined,  being 
apparently  seen  through  a  thin,  luminous  veil,  which  gradually 
extends  over  the  spot.     The  process  sometimes  eventuates  in  the 
sudden  appearance  of  a  luminous  line  traversing  the  dark  inter- 
val, which  is  then  rapidly  followed  by  the  filling  up  and  disap- 
pearance of  the  spot. 

The  velocity  of  the  inward  movement  of  the  penumbral  edges 
is  found  to  have  exceeded,  in  some  of  the  larger  spots,  44  miles 
per  hour. 

274.  Periodicity  of  the  Spots.     It  has  been  ascertained, 
by  systematic  observations  upon  the  spots,  that  their  number 
varies  considerably  in  different  years.     It  will  sometimes  happen 
that,  on  every  clear  day  during  a  particular  year,  the  sun's  disc 
always  contains  one  or  more  of  them,  while,  in  another  year,  for 
weeks  or  even  months  together,  no  spots  of  any  kind  can  be  per- 
ceived.    After  twenty-five  years  of  continued  observations,  M. 
Schwabe  discovered  that  there  was  a  regular  alternate  increase 
and  decrease  in  the  varying  numbers  and  sizes  of  the  spots  ob- 


166  THE  SUN  AND  ATTENDANT  PHENOMENA. 

served  during  successive  years ;  the  period  from  one  maximum 
or  one  minimum  to  another  being  about  ten  years.  More  recently, 
Prof.  Wolf,  of  Zurich,  by  a  careful  discussion  of  the  observations 
of  the  solar  spots  made  'during  the  last  one  hundred  years,  has 
determined  that  the  period  of  the  spots  has  varied,  during  this 
interval,  from  8  to  16  years,  and  that  its  mean  value  has  been 
11.11  years.  1860  was  the  last  year  of  maximum.  Agreeably 
to  the  mean  period,  the  year  1866  should  have  been  a  year  of 
minimum  spots. 

275.  Faculae.     Curved  or  branching  streaks  more  luminous 
than  the  general  body  of  the  sun,  are   frequently  perceived 
upon  parts  of  his  disc,  especially  in  the  region  of  large  spots, 
or  of  extensive  groups  of  spots,  or  in  localities  where  dark  spots 
subsequently  make  their  appearance.     These  are  called  Faculce. 
They  are  chiefly  to  be  seen  near  the  margin  of  the  disc.     Adja- 
cent bright  spaces  are  also  an  invariable  accompaniment  of  the 
spots.     These,  in  the  majority  of  instances,  are  most  conspicuous 
behind  the  spots,  or  in  a  direction  opposite  to  that  of  the  sun's 
rotation. 

It  has  recently  been  established  by  an  observation  made  by 
Dawes,  a  distinguished  English  astronomer,  that  the  faculce  are 
ridges  or  masses  of  luminous  matter,  elevated  above  the  general  level 
of  the  sun's  surface.  In  1859,  he  observed  a  bright  streak  at  the 
very  edge  of  the  disc,  which  projected  irregularly  beyond  the 
circular  contour  of  the  edge,  like  a  low  range  of  hills.  For  such 
elevations  to  have  been  distinctly  perceptible,  their  actual  height 
could  not  have  been  less  than  500  miles,  and  was  probably  two 
or  three  times  as  great  as  this. 

276.  General   Telescopic  Appearance  of  Sun's  Disc. 
The  part  of  the  sun's  disc  not  occupied  by  spots  is  far  from  being 
uniformly  bright.     Inequalities  of  brightness  prevail  in  all  parts 
of  the  disc,  which  give  it  a  coarsely  mottled  appearance.     When 
more  attentively  scrutinized,  its  grou-nd  is  seen  to  be  finely  mot- 
tled with  minute  dark  dots  or  pores,  which  often  appear  to  be  in 
a  state  of  change.     It  is  also  observed  that  the  general  luminous 
surface  of  the  sun  presents  the  appearance  of  bright  granules 
scattered  irregularly  over  it,  and  that,  on  the  darker  spaces  be- 
tween the  granulated  portions,  the  minute  dark  pores  are  espe- 
cially prevalent.     It  is  not  yet  decided  whether  these  bright 
granules  are  to  be  regarded  as  distinct  masses  of  greater  bright- 
ness, or  as  merely  different  conditions  of  the  luminous  cloudy 
surface,  diversified  by  elevated  ridges  or  waves.     This  general 
granulation  of  the  surface  is  entirely  wanting  on  the  facula3,  and 
on  the  luminous  border  of  the  penumbra  of  each  of  the  dark 
spots.     But  lines  of  distinct,  elongated,  and  comparatively  bright 
masses  are  often  seen  projected  on  the  penumbra,  directed  toward 
the  centre  of  the  spot,  and  even  extending  irregularly  into  the 
umbra,  or  central  black  spot. 


PERIOD    OF   ROTATION.  167 

277.    Hotioii*    of   the    Spots :— Rotation    of    the    Sun. 

When  the  positions  of  the  spots  on  the  disc  are  observed  from 
day  to  day,  it  is  perceived  that  they  all  have  a  common  motion 
in  a  direction  from  east  to  west.  Some  of  the  spots  close  up  and 
vanish  before  they  reach  the  western  limb ;  others  disappear  at 
the  western  limb,  and  are  never  afterwards  seen ;  a  few,  after 
becoming  visible  at  the  eastern  limb,  have  been  seen  to  pass 
entirely  across  the  disc,  disappear  from  view  at  the  western  limb, 
and  reappear  again  at  the  eastern  limb.  The  time  employed  by 
a  spot  in  traversing  the  sun's  disc  is  about  14  days.  About  the 
same  time  is  occupied  in  passing  from  the  western  to  the  eastern 
limb,  while  it  is  invisible.  The  motions  of  the  spots  are  account- 
ed for,  in  all  their  circumstances,  by  supposing  that  the  sun  has 
a  motion  of  rotation  from  west  to  east,  around  an  axis  nearly 
perpendicular  to  the  plane  of  the  ecliptic;  and  that  the  spots  are 
portions  of  the  solid  body  of  the  sun.  The  truth  of  this  explana- 
tion of  the  apparent  motions  of  the  sun's  spots,  is  confirmed  by 
the  changes  which  are  observed  to  take  place  in  the  magnitude 
and  form  of  the  more  permanent  spots  during  their  passage  across 
the  disc.  When  they  first  come  into  view  at  the  eastern  limb, 
they  appear  as  a  narrow  dark  streak.  As  they  advance  towards 
the  middle  of  the  disc,  they  gradually  open  out  and  increase  in 
magnitude ;  and  after  they  have  passed  the  middle  of  the  disc, 
contract  by  the  same  degrees  until  they  are  again  seen  as  a  mere 
dark  line  upon  the  western  limb. 

A  spot  returns  to  the  same  position  on  the  disc  in  about  27^ 
days.  This  is  not,  however,  the  precise  period  of  the  sun's  rota- 
tion ;  for  during  this  interval  the  sun  has  apparently  moved 
forward  nearly  a  sign  in  the  ecliptic ;  the  spot  will  therefore  have 
accomplished  that  much  more  than  a  complete  revolution,  when 
it  is  again  seen  by  an  observer  on  the  earth  in  the  same  position 
on  the  disc. 

27§.  Period  of  Rotation.  The  apparent  position  of  a  spot 
with  respect  to  the  sun's  centre  may  be  accurately  determined, 
from  day  to  day,  by  observing,  when  the  sun  is  crossing  the 
meridian,  the  right  ascension  and  declination  both  of  the  spot 
and  centre.  From  three  or  more  observations  of  this  kind  the 
period  of  the  sun's  rotation  and  the  position  of  his  equator  may 
be  ascertained. 

The  period  of  the  sun's  rotation,  as  determined  from  observa- 
tions upon  tbe  spots,  is  found  to  increase  with  the  latitude  of  the 
spot;  from  which  it  is  to  be  inferred  that  the  spots  are  not  sta- 
tionary, and  have  different  rates  of  motion  along  the  surface,  in  a 
direction  parallel  to  the  equator.  If,  as  seems  most  probable,  the 
general  direction  of  motion  is  opposite  to  that  of  the  rotation,  then 
the  spots  nearest  the  equator  have  the  slowest  motion :  and  the 
period  of  rotation  deduced  from  observations  upon  these  spots  ap- 
proximates most  nearly  to  the  actual  period  of  rotation  of  the  body 


168  THE  SUN  AND  ATTENDANT  PHENOMENA. 

of  the  sun.  The  period  in  question  is  25  days.  Spots  observed 
in  the  latitude  34°  give  a  period  of  nearly  27  days.  The  incli- 
nation of  the  sun's  equator  to  the  ecliptic  is  about  7^°  ;  and  the 
heliocentric  longitude  of  the  ascending  node  of  the  equator  is 
about  740. 

279.  Regions  of  the  Spots.  The  solar  spots  are  mostly 
confined  to  two  zones  parallel  to  the  equator,  and  extending 
from  5°  to  35°  of  latitude.  Beyond  35°  they  are  rarely  seen, 
and  in  the  polar  regions  never.  The  actual  equator  is  also  sel- 
dom, if  ever,  visited  by  spots.  They  are  most  abundant  toward 
the  middle  of  the  spot-belts,  and  prevail  more  in  the  northern 
than  in  the  southern  hemisphere.  It  is  observed  that  the  spots 
have  a  tendency  to  form  groups  lying  in  lines  or  belts  parallel  to 
the  equator.  This  is  apparently  the  result  of  a  tendency  of  new 
spots  to  break  out  behind  the  old  ones. 

2§O.  Nature  of  the  Spots.  The  dark  spots  on  the  sun  are 
depressions  below  the  luminous  surface.  This  important  fact  was 
first  established  by  Dr.  Wilson,  of  Glasgow.  He  noticed  that  as 
a  large  spot,  which  was  seen  in  November,  1769,  came  near  the 
western  limb,  the  penumbra  on  the  side  toward  the  centre  of  the 
disc  contracted  and  disappeared,  and  that  afterwards  the  luminous 
matter  on  that  side  seemed  to  encroach  upon  the  central  black 
spot,  while  in  other  parts  the  penumbra  underwent  but  little 
change.  On  the  reappearance  of  the  spot  at  the  eastern  limb,  he 
found  that  the  penumbra  was  again  wanting  on  the  side  toward 
the  centre  of  the  disc ;  and  that  when  this  part  made  its  appear- 
ance, after  the  spot  had  advanced  a  short  distance  upon  the  disc, 
it  was  much  narrower  than  the  opposite  part.  These  various 
appearances  of  the  spot  in  question  are  represented  in  Fig.  73. 


0 


FIG.  73. 

They  show  conclusively  that  both  the  black  central  spot  and 
the  penumbra  were  below  the  luminous  surface  of  the  sun.  Dr. 
Wilson  estimated  the  depth  of  the  spot  to  be  nearly  4,000  miles. 
It  has  since  been  observed  that  similar  changes  of  appearance 
are  experienced  by  the  spots  in  general,  in  their  passage  across 
the  disc. 

2§1.  Theories  of  Physical  Constitution  of  Sun ;  and  of 
Formation  of  Spots. 

Wilson's  Theory.  Dr.  Wilson  drew  from  the  various  appear- 
ances of  the  spot  observed  by  him  in  1769  the  natural  conclusion 


PHOTOSPHERE  OF  THE  SUIT.  169 

that  the  solar  spots  were  the  dark  body  of  the  sun,  seen  through 
excavations  made  in  the  luminous  matter  at  the  surface.  The 
luminous  matter  he  conceived  to  have  the  consistence  of  a  fog 
or  cloud,'  rather  than  of  a  liquid ;  and  suggested  that  openings 
might  be  made  in  it  by  the  working  of  some  sort  of  elastic  vapor 
generated  within  the  dark  globe.  The  penumbra  surrounding 
each  black  spot  he  conjectured  to  be  the  sloping  sides  of  the 
opening  in  the  stratum  of  luminous  clouds. 

HerscheVs  Theory.  Sir  William  Herschel,  after  an  assiduous 
study  of  the  aspects  and  phenomena  of  the  sun's  spots,  adopted 
substantially  Dr.  Wilson's  views,  but  conceived  it  to  be  necessary 
in  order  to  explain  the  uniform  shade  of  the  penumbra,  to  sup- 
pose the  existence  of  an  opake,  non-luminous,  cloudy  stratum, 
posited  between  the  luminous  medium  and  the  dark  solid  globe. 
On  this  hypothesis,  the  spots  are  accounted  for  by  supposing 
that  openings  occasionally  take  place  in  both  the  luminous  and 
non-luminous  envelopes,  through  which  the  dark  body  of  the 
sun  is  seen.  The  penumbra  is  the  portion  of  the  obscure  enve- 
lope situated  immediately  around  the  opening  made  in  it,  and 
shining  by  reflected  light  only.  Herschel  supposed  the  openings 
to  be  made  by  the  exertion  of  some  sort  of  explosive  energy 
from  beneath ;  and  that  the  same  upheaving  agency,  when  not 
of  sufficient  intensity  to  rend  the  luminous  envelope,  forced  it  up 
into  masses  or  waves  of  hundreds  of  miles  in  height.  The 
ridges  of  these  waves  he  conceived  to  be  the  faculae,  which  are 
distinctly  seen  only  when  near  the  margin  of  the  disc,  because 
the  waves  there  appear  in  profile,  and  when  near  the  middle  of 
the  disc  are  seen  in  front,  or  foreshortened.  Sir  John  Herschel, 
who  has  also  been  an  attentive  observer  of  the  sun's  spots,  has 
advanced  the  opinion  that  the  agency  by  which  the  spots  are 
formed,  is  exerted  from  above  downwards,  instead  of  from  below 
upwards. 

Kecent  observations  made  by  Dawes  indicate  the  existence  of 
a  second  non-luminous  envelope,  posited  below  that  which  is 
seen  in  the  penumbra  of  a  spot ;  that  this  is  seen  in  the  umbra, 
and  that  the  black  nucleus  often  observed  near  the  centre  of  the 
umbra  is  an  opening  made  in  this  lower  stratum. 

2§2.  Photosphere  of  the  Sun.  It  is  the  received  opinion 
among  astronomers  of  the  present  day,  that  the  sun,  as  main- 
tained by  Wilson  and  Herschel,  consists  of  a  comparatively  dark 
globe,  either  solid  or  liquid,  surrounded  by  one  or  more  lumi- 
nous envelopes,  in  a  vaporous  or  nebulous  condition,  and  some 
thousands  of  miles  in  total  height.  This  exterior  region  per- 
vaded by  the  medium  which  is  the  great  source  of  the  sun's 
light,  is  called  the  photosphere  of  the  sun.  That  this  luminous  me- 
dium is  really  in  the  aeriform  state,  or  in  the  condition  of  cloudy 
masses  floating  in  a  gaseous  medium,  may  be  inferred  from 
its  great  mobility.  The  velocity  of  expansion  and  contraction 


170  THE  SUN  AND   ATTENDANT  PHENOMENA. 

of  the  spots,  which  often  exceeds  40  miles  per  hour,  is  incom- 
patible with  the  supposition  of  a  liquid  condition. 

It  does  not  necessarily  follow  from  the  fact  that  the  solid  or 
liquid  globe  appears  perfectly  dark,  that  it  has  no  degree  of 
luminosity ;  for  it  has  been  observed  that  intensely  ignited  solids 
appear  only  as  black  spots  on  the  disc  of  the  sun,  when  held 
between  the  sun  and  the  eye.  Professor  Henry,  and  more 
recently  Professor  Secchi,  of  Rome,  has  established  by  experi- 
ment that  the  dark  spots  emit  less  heat  than  the  luminous  sur- 
face. 

But  it  is  to  be  observed  that  the  result  of  the  experiments 
does  not  give  any  decisive  indication  as  to  the  comparative  tem- 
peratures of  the  photosphere  and  dark  body  of  the  sun,  since  a 
considerable  fraction  of  the  heat  radiated  from  the  latter  is  no 
doubt  intercepted  by  the  sun's  atmosphere,  below  the  level  of 
the  luminous  surface. 

Depth  of  photosphere.  Secchi  has  succeeded,  from  measure- 
ments made  upon  several  spots  at  the  time  of  the  disappear- 
ance of  the  penumbra  on  the  side  toward  the  centre  of  the 
disc,  in  effecting  an  approximate  determination  of  the  depth  of 
the  photosphere,  on  the  supposition  that  the  penumbra  is  made 
up  of  the  sloping  sides  of  an  opening  in  a  single  luminous  enve- 
lope (281).  He  estimates  the  depth  to  be  about  one-third  of 
the  radius  of  the  earth,  or  1,300  miles.  M.  Faye  has  undertaken, 
to  determine  the  depth  of  the  photosphere  by  a  different  method, 
and  makes  it  about  4,000  miles. 

283.  Luminous  Appearances  exterior  to  the  Photo- 
sphere.    Whenever  the  sun  becomes  totally  eclipsed  by  the 
moon  interposed  between  it  and  the  eye  of  the  observer,  the 
region  exterior  to  the  photosphere  is  seen  to  be  pervaded,  for  a 
considerable  distance,  by  luminous  matter,  which  offers  a  variety 
of  remarkable   appearances.      The  principal  of  these  are  the 
corona,  luminous  streamers,  or  jets  of  light,  and  rose-colored  pro- 
tuberances. 

284.  The  Corona  is  a  ring  or  halo  of  white  light  encircling 
the  sun,  which  becomes  visible  when  the  body  of  the  sun  is  con- 
cealed from  view.     It  is  brightest  next  the  dark  limb  of  the 
moon,  where  it  has  a  rosy  tint,  and  gradually  decreases  in  lustre 
until  it  becomes  undistinguishable  from  the  general  light  of  the 
sky  (See  Plate  II.).     It  has  presented  to  observers  more  or  less 
of  a  radiated  appearance,  as  if  made  up  of  luminous  radiations, 
or  traversed  by  them.     This  appearance  has  been  less  distinct 
near  the  moon  where  the  corona  is  brightest  than  at  the  more 
distant  and  fainter  portions.     In  the  total  eclipse  of  July  18, 
1860,  the  interruptions  of  continuity  became  distinctly  percep- 
tible at  a  distance  from  the  photosphere  of  the  sun  equal  to  the 
sun's  semi-diarneter,  or  more  than  400,000  miles.     Beyond  this, 
the  corona  was  distinctly  radiated.     Its  extreme  breadth,  both 


HEIGHT  AND  EXTENT  OF   PROTUBERANCES.  171 

in  that  eclipse  and  the  eclipse  of  September  7,  1858,  exceeded 
the  diameter  of  the  sun,  or  850,000  miles.  The  extreme  outline 
of  the  corona  is  perceptibly  elliptical  in  its  form  ;  the  major  axis 
lying  in  the  plane  of  the  sun's  equator. 

285.  The  Luminous  Streamers,  whether  forming  part  of 
the  corona,  or  distinct  from  it,  have  been  seen  to  extend,  at  par- 
ticular points,  far  beyond  the  general  outline  of  the  corona.     In 
the  eclipse  of  1860,  some  of  them  were  traced  to  a  distance  from 
the  sun  s  photosphere  equal  to  twice  the  diameter  of  the  sun,  or 
1,700,000  miles  (Plate  II.).     They  present,  in  general,  the  appear- 
ance of  radiations  of  luminous  matter,  in  directions  perpendicular 
to  the  sun's  surface. 

Observations  made  with  the  polariscope  have  established  that 
the  light  of  the  corona  and  streamers  is  in  part  reflected  light. 

286.  The  Rose-colored  Protuberances  are  regarded  by 
observers  as  the  most  remarkable  and  beautiful  phenomena  wit- 
nessed in  total  solar  eclipses.     They  consist  of  apparent  cloudy 
masses,  more  or  less  tinged  with  red  light,  and  of  various  forms 
and  sizes,  noticed  just  without  the  dark  limb  of  the  moon  (Plate 
II.).     In  some  instances,  they  have  been  seen  entirely  detached 
from  the  moon's  limb.     They  are  seen  at  various  points  of  the 
limb,  and  in  every  variety  of  position  with  respect  to  the  equator 
of  the  sun.     The  latter  circumstance  shows  that  they  have  no 
connection  with  the  sun's  spots,  since  these  do  not  occur  in  high 
latitudes. 

It  has  been  repeatedly  observed  that,  in  the  progress  of  a  total 
eclipse  of  the  sun,  the  protuberances  which  become  visible  at  the 
eastern  limb  of  the  moon  continually  decrease  in  their  apparent 
dimensions,  as  if  the  moon  were  screening  more  and  more  of  them 
from  view ;  while  those  seen  at  the  western  limb  continually  in- 
crease in  their  dimensions,  as  if  they  were  more  and  more  un- 
covered by  the  moon  in  its  advance.  These  facts  indicate  that 
the  protuberances  in  question  are  luminous  masses  connected  with 
the  sun,  and  elevated  above  the  photosphere.  Careful  measurements 
have  fully  established  this  conclusion. 

287.  Height  and  Extent  of  the  Protuberances.  Some  of 
the  protuberances  observed  in  the  eclipse  of  1860,  had  an  appa- 
rent height  of  nearly  70,000  miles.     The  breadth  of  single  pro- 
tuberances is  but  a  few  minutes  of  arc,  but  they  sometimes  extend 
in  a  continuous  chain  for  many  degrees.     Near  the  end  of  the 
eclipse  of  1860,  one  chain  of  low  elevations  was  observed  by 
Secchi,  just  without  that  part  of  the  moon's  contour  at  which  the 
sun  was  about  to  make  its   appearance,  which  extended  60°. 
About  the  middle  of  the  eclipse,  no  less  than  ten  distinct  protu- 
berances were  counted,  which  were  about  regularly  distributed 
around  the  disc.     In  view  of  these  facts,  it  seems  highly  probable, 
as  intimated  by  Secchi,  that  the  rose-colored  protuberances  ob- 
served in  that  eclipse,  were  but  the  higher  portions  of  cloudy 


172  THE  SUN"  AND  ATTENDANT  PHENOMENA. 

masses  that  formed  at  a  lower  level,  one  continuous  reddish  enve- 
lope surrounding  the  sun.  This  envelope  must  have  extended 
upwards  from  the  sun's  photosphere  to  the  height  of  thousands 
of  miles,  and  risen  at  some  points  into  cloudy  peaks  of  tens  of 
thousands  of  miles  in  height. 

2  §8.  Mature  of  the  Corona.  The  corona  is  supposed  by 
Sir  John  Herschel,  and  other  astronomers,  to  be  a  gaseous  solar 
atmosphere,  extending  above  the  sun's  photosphere ;  but  its  vast 
extent  (284)  seems  to  be  fatal  to  this  explanation.  Upon  no 
reasonable  supposition  that  can  be  made  with  regard  to  the  effect 
of  the  sun's  heat  in  expanding  such  an  atmosphere,  can  it  be 
supposed  to  have  sufficient  density  to  reflect  a  sensible  amount 
of  light,  beyond  a  few  thousand  miles  from  the  body  of  the  sun. 
The  natural  indications  of  the  phenomena  are  that  the  corona 
consists  of  luminous  matter  streaming  off  from  the  sun  into 
space ;  and  at  the  same  time  that  the  appearance  of  distinct  radi- 
ations arises-from  an  inequality  of  emanation  from  different  parts 
of  the  sun's  surface.  The  elliptical  form  of  the  corona  (284)  in- 
dicates that  the  emission  of  luminous  matter  is  most  abundant 
from  the  equatorial  regions. 

It  has  been  suggested  that  the  radiated  appearance  of  the 
corona  may  result  from  a  partial  interception  of  the  sun's  light 
by  clouds  floating  in  the  sun's  atmosphere ;  but  the  course  of 
the  rays  which  pass  unobstructedly  through  the  spaces  between 
the  clouds,  could  not  be  recognized  unless  they  encounter  matter 
of  sufficient  density  to  reflect  a  sensible  quantity  of  light  to  the 
eye,  and  such  matter  cannot  extend  to  a  distance  of  more  than  a 
million  of  miles  from  the  sun  (285),  unless  there  are  material 
emanations  proceeding  from  the  sun.  We  can  only  avoid  this 
conclusion  by  assuming  that  there  is  a  dense  mass  of  meteoric 
bodies,  or  of  cosmical  matter,  revolving  around  the  sun  within 
this  distance. 

It  is  maintained  by  some  astronomers  that  the  corona  and 
streamers  are  phenomena  of  diffraction,  resulting  from  the  pas- 
sage of  the  sun's  light  near  the  borders  of  the  moon.  But  upon 
this  explanation  there  ought  to  be  certain  attendant  phenomena 
of  variegated  colors  which  are  not  in  reality  seen. 

2§9.  Temperature  of  the  Sun's  Surface.  To  an  observer 
at  the  luminous  surface  of  the  sun,  its  disc  would  appear  to  cover 
an  entire  hemisphere  in  the  heavens,  and  therefore  the  heat  that 
falls  upon  a  small  area  at  the  surface  of  the  sun  should  exceed 
that  received  upon  the  same  area  at  the  distance  of  the  earth,  in 
the  proportion  that  a  hemisphere  of  the  heavens  exceeds  the  area 
occupied  by  the  sun's  disc  as  seen  from  the  earth,  or  nearly  in 
the  ratio  of  100,000  to  1.  A  heat  many  times  less  intense  than 
this  suffices  to  dissipate  the  most  refractory  metals  in  vapor. 

That  the  calorific  rays  emitted  from  the  sun  have  a  far  higher 
intensity  than  those  which  proceed  from  the  hottest  furnaces,  or 


173 

result  from  the  most  vivid  ignition  obtained  by  chemical  or  gal- 
vanic processes,  may  be  inferred  from  the  fact  that  they  penetrate 
glass  with  far  greater  facility. 

Inequalities  of  temperature.  Secchi  has  made  a  series  of  obser- 
vations upon  the  comparative  amounts  of  heat  received  from 
different  parts  of  the  sun's  surface.  He  finds  that  the  polar  emit 
leas  heat  than  the  equatorial  regions ;  and  that  the  two  hemi- 
spheres separated  by  the  equator  have  not  exactly  the  same  tem- 
perature. It  also  appears  from  his  observations  that  the  breaking 
out  of  a  spot  at  any  point  of  the  sun's  disc,  occasions  a  fall  of 
temperature  there  and  at  surrounding  points ;  and  that  the  faculse 
do  not  sensibly  augment  the  temperature  of  the  points  where 
they  make  their  appearance. 

Also,  the  calorific  rays  proceeding  from  the  centre  of  the  sun's 
disc  have  a  higher  intensity  than  those  proceeding  from  the  bor- 
ders. The  same  is  true  of  the  luminous  rays.  From  this  fact  it 
is  inferred  that  the  sun  is  surrounded  by  an  atmosphere  extend- 
ing far  above  its  photosphere,  or  by  some  form  of  matter,  in  a 
condition  to  intercept  a  large  amount  of  light  and  heat. 

290.  Intensity  of  Sun's  Light.     The  most  intense  artificial 
lights  are  the  Drummond  Light,  produced  by  the  flame  of  the 
oxyhydrogen  lamp  directed  against  a  surface  of  chalk,  and  the 
Electric  Light,  generated  by  the  passage  of  a  galvanic  current 
between  two  charcoal  points.     Fizeau  and  Foucault  found,  by 
ingenious  and  carefully  conducted  experiments,  that  the  light  of 
the  sun's  disc  exceeded  in  intensity  the  Drummond  light,  in  the 
ratio  of  146  to  1;  and  that  it  exceeded  the  electric  light  from 
forty  large  plates  of  a  Bunsen's  battery  in  the  ratio  2|  to  1.     It 
appears,  therefore,   that   the   electric   light  is   the   only   artifi- 
cial  light  that   approximates  in   intensity  to  the  light  of  the 
sun. 

291.  Origin  of  the  Sun's  Heat.     There  would  seem  to  be 
but  two  possible  physical  causes  in  operation  that  might  be  ade- 
quate to  the  development  and  maintenance  of  the  high  tempera- 
ture of  the  sun.     These  are — 

(1.)  The  contraction  of  the  body  of  the  sun  from  an  original 
vaporous  state  to  its  present  size  and  density. 

(2.)  The  fall  of  meteoric  masses  into  the  photosphere  of  the 
sun. 

Upon  the  hypothesis,  hereafter  considered,  that  comets  and 
meteors  were  originally  discharged  from  the  surface  of  the  sun 
during  the  successive  stages  of  vaporous  diffusion  through  which 
the  sun's  mass  is  supposed  to  have  passed,  these  two  causes  are, 
physically  speaking,  essentially  the  same;  since  the  heat  •ulti- 
mately developed  must  be  the  same,  whether  the  subsidence  of 
the  matter  is  by  gradual  contraction,  or  by  gravitation.  Yet  the 
fall  of  the  revolving  meteors  one  after  another  into  the  photo- 
sphere of  the  sun  might  now  determine  a  much  higher  tempera- 


174  THE  SUN   AND  ATTENDANT  PHENOMENA. 

ture  than  would  have  resulted  from  contraction  alone.  The  heat 
due  to  their  fall  has  been,  as  it  were,  stored  up  in  these  meteoric 
bodies,  to  be  suddenly  developed,  instead  of  being  gradually 
dissipated  during  the  ages  in  which  the  sun  has  been  going 
through  its  process  of  formation. 

292.  Results  of  Recent  Investigations,  concerning  the 
sun's  spots  and  physical  constitution.  The  careful  scrutiny  and 
assiduous  study,  by  several  astronomers,  of  all  the  phenomena 
observable  on  the  surface  of  the  sun  for  a  number  of  years  past, 
has  led  to  the  following  important  discovery  : 

The  sun's  spots  are  for  the  most  part  developed  by,  or  in  some 
way  connected  with,  the  operation  of  a  physical  agency  exerted  by 
the  planets  upon  the  photosphere.  This  remarkable  fact  has  been 
conclusively  established  by  the  observations  of  Schwabe,  Car- 
rington,  Secchi,  and  others ;  and  especially  by  the  detailed  dis- 
cussion to  which  all  the  reliable  observations  upon  the  spots, 
made  during  the  last  100  years,  have  been  subjected  by  Professor 
"Wolf,  of  Zurich.  The  planets  which  exercise  the  greatest  influ- 
ence are  Jupiter  and  Venus.  The  planetary  agency  is  directly 
recognized  in  the  origination  of  spots  on  the  parts  of  the  sun's 
surface  brought  by  the  rotation  into  favorable  positions,  and  in 
the  subsequent  changes  experienced  by  the  spots  while  subject 
to  the  direct  action  of  the  planet.  It  is  also  shown  by  the 
dependence  of  the  epochs  of  the  maximum  and  minimum  of 
spots  upon  the  positions  of  the  planets,  especially  of  Jupiter  and 
Venus. 

Effects  of  so  marked  a  character,  exerted  by  the  planets  upon 
the  photosphere  of  the  sun,  cannot  reasonably  be  attributed  to 
their  natural  attractive  action,  and  must  apparently  result  from 
a  repulsive  or  impulsive  action  exerted  upon  the  photospheric 
matter. 

Rotation  of  Spots.  Some  spots  have  been  observed  to  have  a 
motion  of  rotation  around  their  centres  ;  but  according  to  Dawes, 
who  has  been  a  diligent  observer  of  solar  phenomena  for  many 
years,  this  is  a  phenomenon  of  exceedingly  rare  occurrence  in  the 
case  of  well-developed  spots. 

293.  Theory  of  the  Origin  of  the  Suit's  Spots.  The  follow- 
ing is  a  brief  outliue  of  a  theory  of  the  development  of  the  sun's  spots,  based  upon 
the  principle  of  planetary  action  above  stated. 

(1.)  The  matter  of  the  sun's  photosphere,  and  for  a  certain  distance  beyond  the 
luminous  surface  of  the  photosphere,  is,  either  wholly  or  in  part,  in  a  magnetized 
state,,  and  arranged  in  columns,  or  lines  of  magnetic  polarization,  like  the  auroral 
matter  in  the  upper  atmosphere  of  the  earth. 

(2.)  A  repulsive  or  impulsive  action  exerted  by  the  planets  upon  the  molecules 
of  these  columns,  tends  to  disturb  their  electric  and  magnetic  equilibrium,  and 
induce  electric  discharges  along  certain  of  the  upper  columns,  by  which  they  are 
widely  dispersed,  and  the  mechanical  equilibrium  of  the  portions  below  disturbed, 
lu  this  way,  a  vast  column  of  expanding  and  ascending  matter  is  originated  in  the 
photosphere,  which  in  the  process  becomes  more  or  less  dissipated,  and  may  reveal 
the  body  of  the  sun  to  view. 


ZODIACAL  LIGHT.  175 

(3.)  The  matter  dispersed,  from  a  loss  of  magnetic  intensity,  or  by  the  electriV 
discharges,  and  certain  portions  of  the  vaporous  matter  of  the  column,  as  they  risb 
above  the  photosphere,  are  brought  into  that  subtle  or  nebulous  condition  observed 
in  the  matter  of  cornets,  in  which  it  becomes  subject  to  an  effective  repulsion 
from  the  sun,  and  so  is  expelled  indefinitely  into  space.  Other  portions  may  be- 
come condensed  above  the  photosphere,  and  subside  into  it. 

(4.)  The  planets  may  be  conceived  to  operate  in  two  ways,  to  initiate  the  process 
of  dispersion  of  the  tops  of  the  photospheric  columns,  and  so  develop  spots  on  the 
sun ;  viz.,  by  originating  in  the  upper  photosphere  electric  currents  radiating  in 
all  directions  from  the  region  exposed  to  most  direct  action,  or  by  developing 
electro-magnetic  currents  running  in  a  direction  opposite  to  that  of  the  rotation. 
Such  radial  electric  currents  would  be  attended  with  an  exaltation  of  the  statical 
electric  condition  of  the  region  exposed  to  planetary  action ;  and  such  magnetic 
currents  would  tend  to  demagnetize,  or  magnetize,  the  upper  photospheric  columns, 
according  as  the  upper  or  lower  currents  prevail. 

It  appears,  from  the  results  of  observation,  that  the  planets  operate  unequally 
in  different  parts  of  the  ecliptic,  and  in  different  relative  positions ;  and  their 
effects  are  apparently  modified,  in  certain  positions,  by  the  electro-magnetic  cur- 
rents developed  in  the  sun's  photosphere  by  the  motion  of  the  solar  system  through 
space. 

(5.)  The  spots  are  more  likely  to  occur  in  low  than  in  high  latitudes,  because  the 
induced  magnetism  of  the  photospheric  columns  has  a  lower  intensity  in  proportion 
as  the  magnetic  latitude  is  less ;  and  spots  do  not  make  their  appearance  on  the 
equator,  nor  in  its  immediate  vicinity,  because  the  columns  of  magnetic  matter  are 
there  parallel  to  the  surface  of  the  sun. 

(6.)  In  the  supposed  electric  discharges  along  the  magnetic  columns,  with  the 
attendant  accumulation  of  luminous  matter  in  certain  localities  above  the  ordinary 
surface  of  the  photosphere,  we  have  at  the.  same  time  an  adequate  explanation  of 
the  faculcK,  and  of  the  rose-colored  protuberances  at  a  still  higher  level. 

When,  in  special  localities,  the  discharge  has  attained  to  a  sufficient  intensity, 
or  continued  for  a  sufficient  length  of  time,  openings  are  made  through  the  whole 
depth  of  the  photosphere,  and  spots  are  seen  in  the  region  where  the  faculce  were 
before  observed. 

It  may  be  added,  in  this  connection,  that  the  supposition  of  the  distribution  of 
the  photospheric  matter  in  separate  columnar  masses,  accords  with  the  granulated 
appearance  presented  by  the  sun's  disc  (276). 

(7.)  The  photospheric  matter  dispersed  by  reason  of  the  varying  action  of  the 
planets,  sufficiently  to  become  subject  to  a  repulsive  action  from  the  sun  as  it 
Hows  away  into  space,  forms  the  corona,  with  its  accompanying  radiations  and 
streamers,  visible  in  total  eclipses. 

.  (8.)  A  portion  of  the  attenuated  matter  thus  expelled  to  an  indefinite  distance 
from  the  sun,  is  received  into  the  upper  atmosphere  of  the  earth,  and,  by  develop- 
ing electric  currents  there,  becomes  one  of  the  operating  causes  of  the  disturbances 
of  the  magnetic  needle  on  the  earth's  surface ;  which  are  observed  to  increase  and 
decrease,  pari  passu  with  the  sun's  spots.  The  impulses  attendant  upon  the 
electric  discharges  occurring  in  the  sun's  photosphere,  and  propagated  indefinitely 
into  space,  constitute  another  cause  of  magnetic  disturbance  upon  the  earth. 

The  escaping  solar  matter  received  into  the  earth's  upper  atmosphere,  supplies 
the  matter  of  terrestrial  auroras,  which  also  have  the  same  periods  as  the  sun's 
spots.  (For  a  more  complete  exposition  of  the  author's  theoretical  views,  see  Am. 
Journal  of  Science,  Yol  XLL,  Nos.  121  and  122.  See  also  Note  in  Appendix.) 


ZODIACAL  LIGHT. 

294.  At  certain  periods  of  the  year  a  luminous  appearance 
is  observed  in  connection  with  the  sun,  extending  upwards  from 
the  western  horizon  after  evening  twilight,  and  from  the  eastern 
horizon  before  daybreak,  which  is  called  the  Zodiacal  Light,  from 
the  circumstance  of  its  being  mostly  comprehended  within  the 


176 


THE   SUN  AND  ATTENDANT  PHENOMENA. 


FiG.    74. 


zodiac.     Its  color  is  white,  with  a  decided  tinge  of  yellow  at  the 

lower  altitudes.  When  most  con- 
spicuous, it  has  a  striking  bril- 
liancy near  the  horizon,  and  fades 
upwards  by  imperceptible  de- 
grees. Its  apparent  form  is  near- 
ly triangular,  the  base  resting  on 
the  horizon,  from  which  it  tapers 
upwards  to  an  indistinct  vertex. 
The  axis,  or  central  line,  lies  near- 
ly in  the  ecliptic.  Its  length 
varies  with  the  season  of  the 
year,  and  the  state  of  the  atmo- 
sphere. As  estimated  from  the 
sun,  it  is  sometimes  more  than 
100°,  but  ordinarily  not  more 
than  40°  or  50°.  Its  breadth 
near  the  horizon  varies  from  S»° 
to  30°  or  40°.  It  is  nowhere 
abruptly  terminated,  but  gradu- 
ally merges  into  the  general  light 
of  the  sky  (Fig.  74). 
295.  It  varie§  in  Distinctness.  Tne  Zodiacal  Light  is 
seen  most  distinctly,  in  our  northern  latitudes,  in  February  and 
March  after  sunset,  and  in  October  and  November  before  sunrise. 
During  the  month  of  March  it  may  be  seen  directed  towards  the 
star  Aldebaran.  In  December,  though  fainter,  it  may  often  be 
seen  both  in  the  morning  and  evening.  Also,  towards  the  sum- 
mer solstice,  it  is  discernible,  in  a  very  pure  state  of  the  atmo- 
sphere, both  in  the  morning  and  evening.  The  reason  of  the 
variations  in  the  distinctness  of  the  zodiacal  light,  from  one  sea- 
son to  another,  is  found  in  the  change  of  its  inclination  to  the 
horizon  at  the  time  of  sunset  or  sunrise,  together  with  the  varia- 
tion that  occurs  in  the  duration  of  twilight.  As  its  length  lies 
nearly  in  the  plane  of  the  ecliptic,  its  inclination  to  the  horizon, 
will  be  different,  like  that  of  this  plane,  according  to  the  differ- 
ent positions  of  the  sun  in  the  ecliptic.  At  sunset,  the  zodiacal 
light  will  be  most  inclined  to  the  horizon,  and  therefore  extend 
farthest  up  in  the  heavens,  towards  the  vernal  equinox,  when 
the  ecliptic  is  at  sunset  most  nearly  perpendicular  to  the  horizon ; 
and  at  sunrise  it  will  be  most  inclined  to  the  horizon  towards  the 
autumnal  equinox,  when  the  inclination  of  the  ecliptic  to  the 
horizon  at  sunrise  is  the  greatest.  The  zodiacal  light  is  much 
brighter  and  more  frequently  observed  between  the  tropics  than 
in  these  latitudes ;  because  the  ecliptic,  in  general,  makes  there  a 
larger  angle  with  the  horizon,  and  twilight  is  of  shorter  dura- 
tion. 

According  to  Arago  it  appears,  from  the  entire  series  of  obser- 


ZODIACAL  LIGHT.  177 

vations  at  Paris  and  Geneva,  that  the  Zodiacal  Light  varies  con- 
siderably from  one  year  to  another,  and  that  the  observed 
variations  cannot  result  entirely  from  changes  in  the  transpa- 
rency of  the  atmosphere.  Extraordinary  changes  of  brightness 
and  form,  in  the  course  of  a  single  evening,  have  also  been 
noticed  by  several  observers,  which  were  regarded  as  decided 
indications  of  a  change  in  the  intrinsic  lustre  or  density  of  the 
substance  of  the  Zodiacal  Light.  But  all  such  abrupt  changes 
may  possibly  be  purely  of  atmospheric  origin. 

296.  !1<  «  <  lit  Observations:— Important  Result*.  The 
most  valuable  series  of  observations  extant  on  the  zodiacal  light 
are  those  which  were  made  in  the  years  1853-4-5,  at  various 
latitudes,  from  41°  49'  N  to  53°  28'  S,  by  Chaplain  Jones,  of  the 
U.  S.  Navy ;  and  by  the  same  observer,  in  the  years  1856-7, 
from  the  elevated  station  of  Quito,  very  near  the  equator. 

The  discussion  of  these  observations  has  furnished  the  fol- 
lowing important  results : 

1.  When  the  observer  was  in  a  position  on  either  side  of  the 
plane  of  the  ecliptic,  the  main  body  of  the  Zodiacal  Light  was  on 
the  same  side  of  the  ecliptic  in  the  heavens ;  and  when  he  was 
in  the  plane  of  the  ecliptic,  this  light  was  equally  divided  by  its 
circle  in  the  heavens. 

2.  When  the  observer  was  carried  by  the  earth's  rotation 
rapidly  towards  or  from  the  plane  of  the  ecliptic,  the  change  of 
the  apex  of  the  light,  and  of  the  direction  of  its  boundary  lines, 
was  equally  great,  and  corresponded  to  the  change  of  place. 

3.  As  the  ecliptic  changed  its  position  with  respect  to  the 
horizon,  the  entire  shape  of  the  Zodiacal  Light  became  changed. 

4.  The  entire  luminous  appearance  consisted  of  a  stronger  light 
at  the  central  part,  and  a  much  broader  diffuse  light  extending 
beyond  this  on  either  side,  and  to  a  greater  height.     The  stronger 
passed  by  degrees  into  the  diffuse  light,  and  the  latter  also  gradu- 
ally faded  away.     Yet  there  was  a  discernible  line  of  greater 
suddenness  of  transition,  that  could  be  taken  for  the  boundary  of 
the  former. 

5.  The  light  was  visible,  with  more  or  less  distinctness,  on. 
every  favorable  night  during  the  entire  period  of  the  observations. 
The  position  of  the  observer,  generally  at  sea,  or  in  the  lower 
latitudes  when  on  land,  was  more  favorable  than  that  which  most 
observers  have  had. 

6.  On  favorable  nights,  when  the  ecliptic  was  nearly  perpen- 
dicular to  the  horizon,  at  the  observers  station,  the  Zodiacal 
Light  was  visible  at  midnight,  over  both  the  western  and  eastern 
horizons.     This  singular  phenomenon  was  observed  at  sea,  at 
certain  stations  within  the  tropics.     At  Quito,  the  light  was  seen 
every  favorable  night,  and  at  all  hours,  to  extend  as  a  broad 
Luminous  Arch,  entirely  from  one  horizon  to  the  other.     At 
midnight  it  had  a  pale  and  nearly  uniform  white  lustre,  from 

12 


178  THE  SUN  AND  ATTENDANT  PHENOMENA. 

one  horizon  to  the  other.     The  breadth  was  then  nearly  uniform, 
and  about  30°. 

297.  Explanation.  No  generally  received  explanation  of 
this  singular  phenomenon  has  yet  been  given.  It  was  at  one 
time  supposed  to  be  the  atmosphere  of  the  sun,  but  Laplace  has 
shown  that  this  explanation  is  at  variance  with  the  theory  of 
gravitation.  He  found  that  at  the  distance  of  about  sixteen  mil- 
lions of  miles  from  the  centre  of  the  sun,  the  centrifugal  force 
due  to  the  sun's  rotation  balanced  the  gravity,  and  therefore  that 
the  solar  atmosphere  could  not  extend  beyond  this;  but  this 
distance  is  less  than  half  the  distance  of  Mercury  from  the  sun, 
whereas  the  substance  of  the  Zodiacal  Light  extends  beyond  the 
orbit  of  Venus,  and  even  beyond  the  earth's  orbit.  The  most 
plausible  theory  of  the  Zodiacal  Light  that  has  been  advanced  is 
that  propounded  by  Laplace,  that  it  consists  of  a  broad  ring,  or 
lenticular  mass  of  nebulous  matter,  encircling  the  sun  in  the 
plane  of  his  equator.  He  supposed  it  to  be  maintained  in  a  per- 
manent condition  by  the  revolution  of  its  particles  around  the  sun. 

But,  as  intimated  ill  former  editions  of  this  work,  another  conception  of  the  me- 
chanical condition  of  such  a  mass  of  nebulous  matter  may  be  formed,  that  accords 
equally  well  with  the  phenomena  of  the  zodiacal  light.  We  may  regard  the  whole 
mass  as  made  up  of  the  streams  of  particles  which  we  have  recognised  as  continu- 
ally in  the  act  of  flowing  away  from  the  sun  (288  and  293),  under  the  operation  of 
a  force  of  solar  repulsion ;  or,  in  other  words,  that  it  is  the  indefinite  continuation 
of  the  corona  observed  in  total  eclipses  of  the  sun,  with  its  attendant  streamers. 
Upon  this  view  it  should  appear  elongated,  like  the  faint  outer  boundary  of  the 
corona  (284),  in  the  plane  of  the  sun's  equator;  and  this  elongation  may  be  attri- 
buted to  a  more  copious  discharge  of  photospheric  matter  from  the  equatorial  than 
from  the  polar  regions  of  the  sun.  Or  we  may  conceive,  in  accordance  with  the 
theoretical  views  of  the  probable  condition  of  the  sun's  photosphere  that  have  been 
presented  (293),  that  the  discharges  may  take  place  from  the  tops  of  the  photo- 
spheric  columns,  in  the  direction  of  their  prolongation.  All  such  discharged  parti- 
cles would  thus  receive  a  projectile  velocity  oblique  to  the  sun's  surface,  and 
toward  the  plane  of  the  equator,  and,  being  subsequently  repelled  by  the  suu, 
would  move  away  into  space  in  hyperbolic  orbits,  convex  toward  the  sun.  As  a 
necessary  consequence,  there  would  be  an  augmentation  of  the  quantity  of  escap- 
ing matter  in  the  plane  of  the  sun's  equator,  and  an  elongation  of  its  visible  por- 
tion in  this  planer'  The  light  may  vary  in  brightness  from  one  year  to  another, 
with  the  varying  activity  of  discharge  from  the  sun's  surface. 

The  appearance  and  phenomena  of  the  Zodiacal  Light  indicate  that  the  principal 
portion  of  the  light  received  experiences  specular  reflection  from  the  particles  of  its 
substance.  In  the  report  of  the  observations  referred  to  in  the  last  article,  this 
idea  is  presented  and  advocated.  Upon  this  view,  the  amount  of  light  reflected 
to  the  eye  from  different  directions  will  increase  with  the  angle  included  between  the 
directions  of  the  incident  and  reflected  rays,  and  with  the  density  of  the  substance. 

The  lateral  shiftings  of  position  of  tlie  light,  as  the  distance  of  the  zenith  from  the 
ecliptic  varies,  may  be  satisfactorily  explained  by  means  of  the  following  general 
considerations : 

1.  By  reason  of  the  small  dimensions  of  the  earth,  as  compared  with  the  vast 
extent  of  the  entire  collection  of  matter  flowing  away  from  the  sun  to  an  indefinite 
distance,  the  density  of  this  nebulous  matter  must  be  sensibly  the  same  for  consi- 
derable distances  from  the  earth,  in  all  directions.  But  beyond  a  certain  distance, 
which  we  wih1  call  D,  the  density  must  begin  sensibly  to  decrease  as  the  line  of 
eight  makes  a  greater  angle  with  the  plane  of  the  ecliptic  (which  is  nearly  coinci- 
dent with  the  plane  of  the  sun's  equator,  the  supposed  plane  of  greatest  density  of 
the  solar  emanations).  This  decrease  of  density  will  be  more  rapid  as  the  portion 


ZODIACAL  LIGHT.  179 

of  nebulous  matter  considered  is  more  remote  from  the  earth.  It  is  plain,  then,  that 
the  light  reflected  to  the  earth,  from  all  portions  of  this  matter  situated  at  distances 
from  the  earth  greater  than  D,  will  decrease  in  intensity  from  the  ecliptic,  in  both 
directions.  The  brightness  of  the  light,  at  its  different  points,  will  also  augment 
as  the  angular  distance  from  the  sun  diminishes.  The  entire  result  from  the  light 
thus  reflected  should  then  be  a  luminous  appearance  similar  to  the  observed  Zodia- 
cal Light,  with  its  axis  or  line  of  greatest  brightness  lying  in  the  ecliptic. 

If  we  now  take  account  of  the  absorptive  action  of  the  atmosphere  upon  the 
transmitted  light,  which  increases  in  proportion  as  the  direction  of  the  ray  makes 
a  less  angle  with  the  plane  of  the  horizon  we  perceive  that  the  axis  will  be  thrown 
to  that  side  of  the  ecliptic  on  which  the  zenith  lies,  whenever  the  inclination  of  the 
ecliptic  to  the  horizon  is  less  than  90°. 

The  light  received  from  all  portions  of  the  nebulous  matter  which  are  at  a  less 
distance  than  D  from  the  earth,  will  produce  a  different  result.  Its  intensity  will 
be  the  same  for  all  points  at  the  same  angular  distance  from  the  sun ;  that" is,  if 
we  disregard  atmospheric  absorption.  If  this  be  taken  into  account,  it  will  be  seen 
that  the  actual  appearance  will  be  a  diffuse  luminosity,  decreasing  in  both  direc- 
tions from  the  vertical  circle  passing  through  the  sun  below  the  horizon.  From  the 
station  of  the  observer,  within  the  shadow  of  the  earth,  the  nearer  portions  of  the 
shining  nebulous  matter  will  lie  in  the  direction  of  this  vertical  circle.  This  cir- 
cumstance will  tend  to  augment  the  brightness  along  thig  circle,  and  in  its  vicinity, 
as  compared  with  points  at  a  distance  from  it. 

The  Zodiacal  Light  observed  is  the  result  of  the  combination  of  this  light,  which 
has  its  axis  in  the  vertical  circle  through  the  sun,  in  its  position  below  the 
horizon,  with  the  stronger  light  reflected  from  the  more  remote  regions,  whose 
axis  lies  in  the  ecliptic. 

2.  The  axes  of  these  two  different  luminosities  coincide  whenever  the  ecliptic  is 
perpendicular  to  the  horizon ;  but  when  these  circles  are  inclined  to  each  other,  a 
greater  portion  of  the  compound  light  falls  on  the  side  of  the  ecliptic  on  which  the 
zenith  and  the  vertical  circle  of  the  sun  lie,  than  on  the  opposite  side.  When  the 
inclination  of  the  circles  increases,  the  disparity  between  the  portions  of  the  light 
that  lie  on  opposite  sides  of  the  ecliptic  becomes  greater,  and  the  axis  and  the 
whole  luminous  appearance  are  displaced  in  the  direction  of  the  vertical  circle. 
This  displacement  should  certainly  take  place  up  to  a  certam  limit  of  increase  in 
the  angle,  which  should  be  greatest  at  the  lower  altitudes,  at  which  the  observed 
displacement  is  greatest.  It  is  also  to  be  noticed  that  there  is  a  secondary- 
cause  in  operation,  tending  to  augment  this  lateral  displacement ;  which  consists 
in  the  fact,  that  as  the  ecliptic  becomes  more  oblique  to  the  horizon,  the  sun  when 
at  the  same  distance  as  before,  along  the  ecliptic  from  the  horizon,  will  be  nearer 
the  horizon  in  the  direction  of  the  vertical  circle,  and  therefore  at  equal  heights 
above  the  horizon,  the  luminosity  which  has  its  axis  in  the  vertical  circle,  will  be 
increased  in  brightness,  and  so  have  a  greater  displacing  effect  on  the  boundary  of 
the  ecliptic  light.  It  will  also  readily  be  perceived  that  up  to  a  certam  amount 
of  deviation  of  the  ecliptic  from  the  vertical  circle,  the  unequal  atmospheric  absorp- 
tion of  the  light  will  operate  to  increase  the  displacement. 

The  Diffuse  Light  noticed  in  the  last  article,  probably  has  its  origin  in  the  portions 
of  the  nebulous  solar  emanation  that  lie  immediately  beyond  the  distance  D,  the 
density  of  which  will  decrease  from  the  ecliptic  more  slowly  than  that  of  the  more 
remote  portions. 

The  Luminous  Arch  seen  at  midnight  in  tropical  regions  (296)  must  be  attributed, 
from  the  present  point  of  view,  to  the  radiant  reflection,  and  feeble  specular  reflec- 
tion at  angles  of  incidence  less  than  45°,  of  the  sun's  rays,  from  the  portion  of  the 
solar  matter  that  extends  indefinitely  beyond  the  earth's  orbit.  The  variations  in 
the  amount  of  light  reflected  from  regions  at  different  angular  distances  from  the 
sun,  in  the  density  of  the  reflecting  substance,  and  in  the  effects  of  atmospheric  ab- 
sorption, tend  to  equalize  the  light  received  from  different  directions  lying  in 
the  plane  of  the  ecliptic.  The  extent  of  the  conical  shadow  cast  by  the  earth  is 
presumably  small  in  comparison  with  that  of  the  nebulous  substance  from  which 
the  light  is  received. 

It  will  be  perceived  that  atmospheric  absorption  plays  the  prominent  part  in  the 
phenomenon  of  the  lateral  displacement  of  the  Zodiacal  Light,  opera  ting  both 
directly  and  indirectly. 


ISO  THE  MOON  AND  ITS  PHENOMENA. 


CHAPTER  XV. 

.    THE  MOON  AND  ITS  PHENOMENA. 
PHASES  OF  THE  MOON. 

298.  THE  most  conspicuous  of  the  phenomena  exhibited  by  the 
moon,  is  the  periodical  change  that  is  observed  to  take  place  in 
the  form  and  size  of  its  disc.  The  different  appearances  which, 
the  disc  presents  are  called  the  Phases  of  the  moon. 

The  phenomenon  in  question  is  a  simple  consequence  of  the 
revolution  of  the  moon  around  the  earth.  Let  E  (Fig.  75) 
represent  the  position  of  the  earth,  ABC  the  orbit  of  the 


O 


o 


FIG.  75. 


moon,  which  we  will  suppose  for  the  present  to  lie  in  the  plane 
of  the  ecliptic,  and  ES  the  direction  of  the  sun.  As  the  distance 
of  the  sun  from  the  earth  is  about  400  times  the  distance  of  the 
moon,  lines  drawn  from  the  sun  to  the  different  parts  of  the 
moon's  orbit,  may  be  considered,  without  material  error,  as  par- 
allel to  each  other.  If  we  regard  the  moon  as  an  opake  non- 
luminous  body,  of  a  spherical  form,  that  hemisphere  which  is 
turned  towards  the  sun  will  be  continually  illuminated,  and 
the  other  will  be  in  the  dark.  Now,  by  virtue  of  the  moon's 
motion,  the  enlightened  hemisphere  is  presented  to  the  earth 
under  every  variety  of  aspect  in  the  course  of  a  synodic  revolu- 
tion of  the  moon.  Thus,  when  the  moon  is  in  conjunction,  as  at 
A,  this  hemisphere  is  turned  entirely  away  from  the  earth,  and 


PHASES  OF  THE  MOOX.  181 

it  is  invisible.  Soon  after  conjunction,  a  portion  of  it  on  the 
right  begins  to  be  seen,  and  as  this  is  comprised  between  the 
right  half  of  the  circle  which  limits  the  vision,  and  the  right  half 
of  the  circle  which  separates  the  enlightened  and  dark  hemi- 
spheres of  the  moon,  called  the  Circle  of  Illumination,  it  will  ob- 
viously present  the  appearance  of  a  crescent,  with  the  horns 
turned  from  the  sun,  as  represented  at  B.  As  the  moon  advances, 
more  and  more  of  the  enlightened  half  becomes  visible,  and  thus 
the  crescent  enlarges,  and  the  eastern  limb  becomes  less  concave. 
At  the  point  C,  90°  distant  from  the  sun,  one-half  of  it  is  seen, 
and  the  disc  is  a  semi-circle,  the  eastern  limb  being  a  right  line. 
Beyond  this  point,  more  than  half  becomes  visible ;  the  nearer 
half  of  the  circle  of  illumination  falls  to  the  left  of  the  moon's 
centre,  as  seen  from  the  earth,  and  thus  becomes  convex  out- 
ward. This  phase  of  the  moon  is  represented  at  D.  When  the 
moon  appears  under  this  shape,  it  is  said  to  be  Gibbous.  In  ad- 
vancing towards  opposition,  the  disc  will  enlarge,  and  the  eastern 
limb  become  continually  more  con  vex  ;  and  finally  at  opposition, 
where  the  whole  illuminated  face  is  seen  from  the  earth,  it  will 
become  a  full  circle.  From  opposition  to  conjunction,  the  nearer 
half  of  the  circle  of  illumination  will  form  the  right  or  western 
limb,  and  this  limb  will  pass  in  the  inverse  order  through  the 
same  variety  of  forms  as  the  eastern  limb  in  the  interval  between 
conjunction  and  opposition.  The  different  phases  are  delineated 
in  the  figure. 

The  moon's  orbit  -is,  in  fact,  somewhat  inclined  to  the  plane 
of  the  ecliptic,  instead  of  lying  in  it,  as  we  have  supposed ;  but, 
it  is  plain  that  its  inclination  cannot  change  the  order,  nor  the 
period  of  the  phases,  and  that  it  can  have  no  other  effect  than  to 
alter  somewhat  the  size  of  the  disc,  at  particular  angular  distances 
from  the  sun.  In  consequence  of  the  smallness  of  the  inclination, 
this  alteration  is  too  slight  to  be  noticed. 

299.  Definitions.     When  the  moon  is  in  conjunction,  it  is 
said  to  be  New  Moon;  and  when  in  opposition,  Full  Moon.     At 
the  time  between  new  and  full  moon,  when  the  difference  of  the 
longitudes  of  the  moon  and  sun  is  90°,  it  is  said  to  be  the  First 
Quarter.     And  at  the  corresponding  time  between  full  and  new 
moon,  it  is  said  to  be  the  Last  Quarter.     In  both  these  positions 
the  moon  appears  as  a  semi-circle,  and  is  said  to  be  dichotomized. 
The  two  positions  of  conjunction  and  opposition  are  called  Syzi- 
gies  ;  and  those  of  the  first  and  last  quarter,  Quadratures.     The 
four  points  midway  between  the  syzigies  and  quadratures  are 
called  Octants. 

300.  Lunar  Month.     The  interval  from  new  moon  to  new 
moon  again,  is  called  a  Lunar  Month,  and  sometimes  a  Lunation. 

The  mean  daily  motion  of  the  sun  in  longitude  is  59'  8".33, 
and  that  of  the  moon  13°  10'  35". 03 ;  wherefore  the  moon  sepa- 
rates from  the  sun  at  the  mean  rate  of  12°  II'  26". 70  per  day ; 


182  THE   MOON   AND   ITS   PHENOMENA. 

and  hence,  to  find  the  mean  length  of  a  lunar  month,  we  have 
the  proportion 

12°  11'  26".70  :  Id.  : :  360°  :  x  =  29d.  12h.  44m.  2.7s. 

3O1.  To  Determine  the  Time  of  Mean  New,  or  Full 
Moon,  in  any  Given  Month.  Let  the  mean  longitude  of  the 
sun,  and  also  the  mean  longitude  of  the  moon,  at  the  beginning  of 
the  year,  be  found,  and  let  the  former  be  subtracted  from  the  latter 
(adding  360°  if  necessary);  the  remainder,  which  call  E,  will  be  the 
mean  distance  of  the  moon  to  the  east  of  the  sun,  at  the  beginning 
of  the  year.  As  the  moon  separates  from  the  sun  at  the  mean  rate 

of  12°  II7  26".70  per  day,  ?0a//TA  wil1  exPress  the  num' 

12     11    26  .70 

ber  of  days  and  fractions  of  a  day,  which  at  this  epoch  have 
elapsed  since  the  last  new  moon.  This  interval  is  called  the 
Astronomical  Epact.  If  we  subtract  it  from  29d.  12h.  44m.  2.7s. 
we  shall  have  the  time  of  mean  new  moon  in  January.  This 
being  known,  the  time  of  mean  new  moon  in  any  other  month 
of  the  year  results  very  readily  from  the  known  length  of  a 
lunar  month. 

The  time  of  mean  new  moon  in  any  month  being  known,  the 
time  of  mean  full  moon  in  the  same  month  is  obtained  by  the 
addition  or  subtraction,  as  the  case  may  be,  of  half  a  lunar 
month. 

This  problem  is  in  practice  most  easily  resolved  with  the  aid 
of  tables.  (See  Problem  XXVII.) 

The  time  of  true  new  moon  differs  from  the  time  of  mean  new 
moon,  for  the  same  reasons  that  the  true  longitudes  of  the  sun 
and  moon  differ  from  the  mean.  The  same  is  true  of  the  time 
of  true  full  moon.  For  the  mode  of  computing  the  time  of  true 
new  or  full  moon  from  that  of  mean  new  or  full  moon,  see 
Problem  XXVII. 

3052.  The  Earth  goes  through  the  same  Phases,  as 
viewed  from  the  moon,  in  the  course  of  a  lunar  month  that  the 
moon  does  to  an  inhabitant  of  the  earth.  But,  at  any  given 
time,  the  phase  of  the  earth  is  just  the  opposite  to  the  phase  of 
the  moon.  About  the  time  of  new  moon,  the  earth,  then  near 
its  full,  reflects  so  much  light  to  the  moon  as  to  render  the  ob- 
scure part  visible.  (See  Fig.  75.) 


MOON'S  RISING,  SETTING,  AND  PASSAGE  OVER  THE 
MERIDIAN. 

3O3.  To  find  the  Time  of   the  Meridian  Passage  of 
the  Moon  011  a  Given  Day. 

Let  S  and  M  denote,  respectively,  the  right  ascension  of  the 

,          sun  and  the  right  ascension  of  the*  moon,  at  noon  on  the  given 

(n         day,  and  m,  s,  the  hourly  variations  of  the  right  ascension  of  the 


PHASES  OF  THE   MOOX.  183 

sun  and  moon  ;  also  let  t  =  the  required  time  of  the  meridian 

passage.  At  the  time  t  the  right  ascensions  will  be, 

for  the  moon M  +  tm, 

for  the  sun S  4-  ts ; 

and,  as  the  moon  is  on  the  meridian,  the  difference  of  these  arcs 

will  be  equal  to  the  hour  angle  t ;  whence, 

t=U  —  S  +  t(m  —  s); 
or,  if  all  the  quantities  be  expressed  in  seconds, 

;=M  —  S+tm  —  s.,  ..(52). 
8600 

Thus,  we  find  for  the  time  of  the  meridian  passage, 

3600  (M-S)          ,,„ 
-3600  —  (m  —  s)" 

The  quantities  M,  S,  ?n,  s,  are,  in  practice,  to  be  taken  from 
ephemerides  of  the  sun  and  moon. 

Example.  "What  was  the  time  of  the  passage  of  the  moon's  centre  over  the  meri- 
dian of  New  York  on  the  1st  of  August,  1837  ? 

When  it  is  noon  at  New  York,  it  is  4h.  56m.  4s.  at  Greenwich.  Now,  by  the 
English  Nautical  Almanac, 

Aug.  1st,  at  4h.  Q)'s  R.  Ascen 8h.  58m.  36.7s. 

"  at5h.     "          "       9       0     38.3 

Ih. :  56m.  4s. : :  2m.   1.6s. :  1m.  53.6s. 

Aug.  1st,  at  4h.  Q)'s  R  Ascen 8h.  58m.  36.7s. 

Variation  of  R.  Ascen.  in  56m.  4s  .  1      53.6 


O's  R.  Ascen.  at  M.  Noon  at  N.  York 9      0     30.3 

Aug.  1st.  O  's  hourly  Variation  of  R.  Ascen 9.704s. 

Ih. :  4h.  56m.  4s.  : :  9.704s. :  47.8s. 

Aug.  Lst,  M.  Noon  at  Greenw.,  O's  R.  Asc. . . .  8h.  45m.  31.5s. 
Variation  of  R.  Ascen.  in  4h.  56m.  4s 47.8 


O's  R.  Ascen.  at  M.  Noon  at  N.  York 8    46      19i3 

Aug.  1st,  M.  Noon  at  Greenw.,  Q)'s  R.  Asc 8h.  50m.  27.7s. 

Aug.  2d,         "  "  ....  9     38      18.7 

24)47      51.0 

Aug.  1st,  f)'s  mean  hourly  Varia.  of  R.  Asc. . .          1     59.6  (m) 
"  O's  "  "  "...  9.7  («) 

m  —  8  =  1     49.9  =  109.98. 
By  Nautical  Almanac,  equation  of  time  =  5m.  59s. 

Ih.  :  5m.  59s. : :  1m.  59.6s. :  11.9s. 

3's  R  Ascen.  at  M.  Noon  at  N.  York 9h.    Om.  30.3s. 

Correction  for  equation  of  time —  11.9 

3'sR.  Ascen.  at  apparent  Noon  at  N.York...  9      0     18.4  (M) 
0'B  ...  8    46     18.3  (S) 

M—  S  =  14       0.1  =  840.1s. 


184  THE  MOON  AND  ITS  PHENOMENA. 

3600 log.  3.55630 

M  —  S  =  840.1   log.  2.92433 

3600  —  (w  —  «)  =  3490.1 ar.  co.  log.  6.45716 

Apparent  time  of  meridian  passage,  14m.  26.5s.  =  866.5s. .  log.  2.93T79 
Equa.  of  time  at  merid.  passage,          5      58 

Mean  time  of  meridian  passage,  Oh.  20m.  24s. 

The  Nautical  Almanac  gives  the  time  of  the  moon's  passage  over  the  meridian 
of  Greenwich  for  every  day  of  the  year.  From  this,  the  time  of  the  passage  across 
the  meridian  of  any  other  place  may  easily  be  determined,  as  follows :  subtract  the 
time  of  the  meridian  passage  at  Greenwich  on  the  given  day,  from  that  on  the  fol- 
lowing day,  and  say,  as  24h.  :  the  difference  : :  the  longitude  of  the  place  :  a  fourth 
term.  The  fourth  term,  added  to  the  time  of  the  meridian  passage  at  Greenwich 
on  the  given  day,  will  give  the  time  of  the  meridian  passage  on  the  same  day  at 
the  given  place. 

304.  Moon's  Rising  and  Setting.     Since  the  moon  has  a 
motion  with  respect  to  the  sun,  the  time  of  its  rising  and  setting 
must  vary  from  day  to  day.     When  first  seen  after  conjunction, 
it  will  set  soon  after  the  sun.     After  this  it  will  set  (at  a  mean) 
about  50m.  later  every  succeeding  night.     At  the  first  quarter, 
it  will  set  about  midnight;  and  at  full  moon,  will  set  about  sun- 
rise, and  rise  about  sunset.     During  this  interval  it  will  rise  in 
the  daytime,  and  all  along  from  sunrise  to  sunset.     From  full 
to  new  moon,  it  will  rise  at  night  and  set  during  the  day ;  and 
the  time  of  the  rising  and  setting  will  be  about  50m.  later  on 
every  succeeding  night  and  day ;  thus,  at  the  last  quarter,  it  will 
rise  about  midnight,  and  set  about  midday. 

305.  Daily  Retardation  of  Moon's  Rising.     The  daily 
retardation  of  the  time  of  the  moon's  rising  is,  as  just  stated,  at  a 
mean,  about  50  minutes  ;  but  it  varies  in  the  course  of  a  revolu- 
tion from  less  than  half  an  hour  to  one  hour,  in  these  latitudes. 
The  retardation  of  the  moon's  rising  at  the  time  of  full  moon, 
varies  from  one  full  moon  to  another,  in  the  course  of  the  year, 
between  the  same  limits.     The  reason  of  these  variations  is  found 
in  the  fact,  that  the  arc  of  the  ecliptic  (12°  ll/)  through  which 
the   moon   moves  away  from   the   sun  in  a  day,  is   variously 
inclined  to  the  horizon,  according  to  its  situation  in  the  ecliptic, 
and  therefore  employs  different  intervals  of  time  in  rising  above 
the  horizon.     This  fact  may  be  very  distinctly  shown  by  means 
of  a  celestial  globe.     It  will  be  seen  that  the  arc  in  question  will 
be  most  oblique  to  the  horizon,  and  rise  in  the  shortest  time, 
in  the  signs  Pisces  and  Aries.     Accordingly,  the  full  moons 
which  occur  in  these  signs  will  rise  with  the  smallest  retarda- 
tion from  day  to  day.     These  full  moons  occur  when  the  sun  is 
in  the  opposite  signs,  Yirgo  and  Libra,  that  is,  in  September  and 
October.     They  are  called,  the  first  the  Harvest  Moon,  and  the 
second  the  Hunter's  Moon.     The  time  of  the  moon's  rising  at 
these  full  moons  will,  for  two  or  three  days,  be  only  about  half 
an  hour  later  than  on  the  preceding  day. 

306.  To  find  the  time  of  the  moon's  rising  or  setting  on  any  given  day.     Compute 


ROTATION  AND  ITERATIONS  OF  THE   MOON.  185 

the  moon's  semi-diurnal  arc  from  equation  (50),  or  (48),  according  as  it  is  the  time 
of  the  apparent  rising  or  setting,  or  the  time  of  the  true  rising  or  setting,  that  is 
desired.  Correct  it  for  the  moon's  change  of  right  ascension  in  the  interval 
between  the  moon's  passage  over  the  meridian  and  setting,  by  the  following  pro- 
portion, 24h  :  24h  +  m  —  *  : :  semi-diurnal  arc  :  corrected  semi-diurnal  arc ;  and 
add  it  to  the  time  of  the  moon's  meridian  passage,  found  as  explained  in  Art.  303. 
The  result  will  be  the  time  of  the  moon's  setting ;  and  if  this  be  subtracted  from 
24  hours,  the  remainder  will  be  the  time  of  the  moon's  rising. 

In  consequence  of  the  change  of  the  moon's  declination  in  the  interval  between 
its  rising  and  setting,  it  would  be  more  accurate  to  compute  the  semi-diurnal  arc 
separately  for  the  moon's  rising.  In  computing  the  semi-diurnal  arc  by  equation 
(48),  the  declination  6  hours  before  or  after  the  meridian  passage  may  be  used  at 
first ;  and  afterwards,  if  a  more  accurate  result  be  desired,  the  calculation  may 
be  repeated  with  the  declination  found  for  the  computed  approximate  time.  In 
equation  (49),  R  =  refraction — parallax  =  34'  54'' — 57'  3"  (at  a  mean)  = 
—  22'  9". 


ROTATION  AND  ITERATIONS  OF  THE  MOON. 

307.  The  moon   presents  continually  nearly  the  same  face 
towards  the  earth  ;  for  the  same  spots  are  always  seen  in  nearly 
the  same  position  upon  the  disc.     It  follows,  therefore,  that  it 
rotates  on  its  axis  in  the  same  direction,  and  with  the  same  angu- 
lar velocity,  or  nearly  so,  that  it  revolves  in  its  orbit,  and  thus 
completes  one  rotation  in  the  same  period  of  time  in  which  it 
accomplishes  a  revolution  in  its  orbit. 

308.  Librations  of  the  JIooii.     The  spots  on  the  moon's 
disc,  although  they  constantly  preserve  very  nearly  the  same 
situations,  are  not,  however,  strictly  stationary.     "When  carefully 
observed,  they  are  seen  alternately  to  approach  and  recede  from 
the  edge.     Those  that  are  very  near  the  edge  successively  disap- 
pear and  again  become  visible.     This  vibratory  motion  of  the 
moon's  spots  is  called  Libration. 

There  are  three  librations  of  the  moon,  that  is,  a  vibratory  motion 
of  its  spots  from  three  distinct  causes. 

(1.)  The  moon's  motion  of  rotation  being  uniform,  small  por- 
tions on  its  east  and  west  sides  alternately  come  into  sight  and 
disappear,  in  consequence  of  its  unequal  motion  in  its  orbit.  The 
periodical  oscillations  of  the  spots  in  an  easterly  and  westerly 
direction  from  this  cause,  are  called  the  Libration  in  Longitude. 

(2.)  The  lunar  spots  have  also  a  small  alternate  motion  from 
north  to  south.  This  is  called  the  Libration  in  Latitude,  and  is 
accounted  for  by  supposing  that  the  moon's  axis  is  not  exactly 
perpendicular  to  the  plane  of  its  orbit,  and  that  it  remains  con- 
tinually parallel  to  itself.  On  this  supposition,  we  ought  some- 
times to  see  beyond  the  north  pole  of  the  moon,  and  sometimes 
beyond  the  south  pole. 

(3.)  Parallax  is  the  cause  of  a  third  libration  of  the  moon.  The 
spectator  upon  the  earth's  surface  being  removed  from  its  centre, 
the  point  towards  which  the  moon  continually  presents  the  same 
hemisphere,  he  will  see  portions  of  the  moon  a  little  different 


186  THE   MOON  AND   ITS  PHENOMENA. 

according  to  its  different  positions  above  the  horizon.  The 
diurnal  motion  of  the  spots  resulting  from  the  parallax,  is  called 
the  Diurnal  or  Parollactic  Libration. 

3O9.  Equator  of  the  Moon.  The  exact  position  of  the 
moon's  equator,  like  that  of  the  sun's,  is  derived  from  accurate 
observations  of  the  situations  of  the  spots  upon  the  disc.  From 
calculations  founded  upon  such  observations,  it  has  been  ascer- 
tained that  the  plane  of  the  moon's  equator  is  constantly  inclined 
to  the  plane  of  the  ecliptic  under  an  angle  of  1°  32',  and  inter- 
sects it  in  a  line  which  is  always  parallel  to  the  line  of  the  nodes. 
It  follows  from  the  last  mentioned  circumstance,  that  if  a  plane 
be  supposed  to  pass  through  the  centre  of  the  moon,  parallel  to 
the  ecliptic,  it  will  intersect  the  plane  of  the  moon's  equator  and 
that  of  its  orbit  in  the  same  line  in  which  these  planes  intersect 
each  other.  The  plane  in  question  will  lie  between  the  plane  of 
the  equator  and  that  of  the  orbit.  It  will  make  with  the  first  an 
angle  of  1°  32',  and  with  the  second  an  angle  of  5°  9'. 


DIMENSIONS  AND  PHYSICAL  CONSTITUTION  OF  THE  MOON. 

31O.  Diameter;—  Surface;—  Volume.  The  phases  of  the 
moon  indicate  that  it  is  an  opake  spherical  body.  Its  diameter 
is  found  by  means  of  equ.  (51),  viz.  : 


in  which  d  will  denote  the  diameter  sought,  E  the  radius  of  the 
earth,  5  the  apparent  diameter  of  the  moon  at  a  given  distance, 
and  H  its  horizontal  parallax  at  the  same  distance. 

The  equatorial  horizontal  parallax  of  the  moon,  at  the  mean 
distance,  is  57'  2".7,  and  its  corresponding  apparent  diameter  is 
31'  7".0:  thus  we  have 


d  =  2R         -  7925-6m-  x  -       =  216L6  miles- 


The  ratio  of  the  diameter  of  the  moon  to  the  mean  diameter 
of  the  earth  (7912.4m.)  is  0.27819.  This  is  very  nearly  equal  to 
•jSj-,  or  a  little  more  than  £.  The  surface  of  the  moon  is  therefore 
to  the  surface  of  the  earth  nearly  as  3a  to  lla,  or  as  1  to  13^; 
and  the  volume  of  the  moon  is  to  the  volume  of  the  earth  nearly 
as  33  to  II8,  or  as  1  to  49. 

311.  Tele§copic  Appearauce§  :—  Inferences.  When  the 
moon  is  viewed  with  a  telescope,  the  edge  of  the  disc  which  bor- 
ders upon  the  dark  portion  of  the  face,  is  seen  to  be  very  irre- 
gular and  serrated  (see  Fig.  76).  It  is  hence  inferred  that  the 
surface  of  the  moon  is  diversified  by  mountains  and  valleys. 
The  truth  of  this  inference  is  confirmed  by  the  fact  that  bright 


PHYSICAL  CONSTITUTION  OF  THE  MOON.  187 

insulated  spots  are  frequently  seen  on  the. dark  part  of  the  face, 
near  the  edge  of  the  disc,  which  gradually  enlarge  until  they 
become  united  to  the  disc.  These  bright  spots  are  doubtless 
the  tops  of  mountains  illuminated  by  the  sun,  while  the  sur- 


FIG.  76. 

rounding  regions  that  are  less  elevated  are  involved  in  darkness. 
The  disc  is  also  diversified  with  spots  of  different  shapes  and 
different  degrees  of  brightness.  The  brighter  parts  are  supposed 
to  be  elevated  land,  and  the  dark  to  be  plains,  and  valleys,  or 
cavities. 

312.  Lunar  Mountains.     The  number  of  the  lunar  moun- 
tains is  very  great.     Many  of  them,  by  their  form  and  grouping, 
furnish  decided  indications  of  a  volcanic  origin. 

From  measurements  inade  with  the  micrometer  of  the  lengths 
of  their  shadows,  or  of  the  distance  of  their  summits  when  first 
illuminated,  from  the  adjacent  boundary  of  the  disc,  the  heights 
of  a  number  of  the  lunar  mountains  have  been  computed.  Ac- 
cording to  Herschel,  the  altitude  of  the  highest  is  only  about  If 
English  miles.  But  Schroeter,  of  Lilienthal,  a  distinguished 
Selenographist,  makes  the  elevation  of  some  of  the  lunar  moun- 
tains to  exceed  5  miles ;  and  the  more  recent  measurements  of 
MM.  Baer  and  Madler,  of  Berlin,  lead  to  similar  results. 

313.  There  are  no  Seas,  nor  other  bodies  of  water,  upon 
the  surface  of  the  moon.     Certain  dark  and  apparently  level 
parts  of  the  moon  were  for  some  time  supposed  to  be  extended 
sheets  of  water,  and,  under  this  idea,  were  named  by  Hevelius 
Mare  Imbriuw,  Mare  Grisium,  etc. :  but  it  appears  that  when  the 
boundary  of  light  and  darkness  falls  upon  these  supposed  seas, 
it  is  still  more  or  less  indented  at  some  points  and  salient  at  others, 
instead  of  being,  as  it  should  be,  one  continuous  regular  curve ; 


188  THE  MOON  AND  ITS  PHENOMENA. 

besides,  when  these  dark  spots  are  viewed  with  good  telescopes, 
they  are  found  to  contain  a  number  of  cavities,  whose  shadows  are 
distinctly  perceived  falling  within  them.  The  spots  in  question 
are  therefore  to  be  regarded  as  extensive  plains  diversified  by 
moderate  elevations  and  depressions.  The  entire  absence  of 
water  also  from  the  farther  hemisphere  of  the  moon  may  be  in- 
ferred from  the  fact  that  the  moon's  face  is  never  obscured  by 
clouds  or  mists. 

3 1 4.  Lunar  Atmosphere.  It  has  long  been  a  question  among  Astrono- 
mers, whether  the  moon  has  an  atmosphere.  It  is  asserted,  that,  if  it  has  any,  it  must 
be  exceedingly  rare,  or  very  limited  in  its  extent,  since  it  does  not  sensibly  dimin- 
ish or  refract  the  light  of  a  star  seen  in  contact  with  the  moon's  limb ;  for  when  a 
star  experiences  an  occultation  by  reason  of  the  interposition  of  the  moon  between 
it  and  the  eye  of  the  observer,  it  does  not  disappear  or  undergo  any  diminution  of 
lustre  until  the  body  of  the  moon  reaches  it ;  and  the  duration  of  the  occultation  is 
as  it  is  computed,  without  making  any  allowance  for  the  refraction  of  a  lunar  atmo- 
sphere.   But  it  is  maintained,  on  the  other  hand,  that  these  facts,  if  allowed,  are 
not  opposed  to  the  supposition  of  the  existence  of  an  atmosphere  of  a  few  miles 
only  in  height ;  and  that  certain  phenomena  which  have  been  observed  afford  in- 
dubitable evidence  of  the  presence  of  a  certain  limited  body  of  air  upon  the  moon's 
surface.     Thus,  the  celebrated  Schroeter,  in  the  course  of  some  delicate  observa- 
tions made  upon  the  crescent  moon,  perceived  a  faint  grayish  light  extending  from 
the  horns  of  the  crescent  a  certain  distance  into  the  dark  part  of  the  moon's  face. 
This  he  conceived  to  be  the  moon's  twilight,  and  hence  inferred  the  existence  of  a 
lunar  atmosphere.     From  the  measurements  which  he  made  of  the  extent  of  this 
light  he  calculated  the  height  of  that  portion  of  the  atmosphere  which  was  capable 
of  affecting  the  light  of  a  star  to  be  about  one  mile.    Again,  iu  total  eclipses  of  the 
sun,  occasioned  by  the  interposition  of  the  moon,  the  dark  body  of  the  moon  has 
been  seen  terminated  by  a  luminous  ring,  which  was  at  first  most  distinct  at  the 
part  where  the  sun  was  last  seen,  and  afterwards  at  the  part  where  the  first  ray 
darted  from  the  sun.    This  is  supposed  to  have  been  a  lunar  twilight.    A  similar 
phenomenon  was  observed  in  the  annular  eclipse  of  1836,  just  before  the  comple- 
tion of  the  ring,  at  the  point  where  the  junction  took  place. 

DESCRIPTION  OF  THE    MOON'S  SURFACE. 

315.  General  Topographical  Features.    The  surface  of  the 
moon,  like  that  of  the  earth,  presents  the  two  general  varieties  of  level  and  moun- 
tainous districts ;  but  it  differs  from  the  earth's  surface  in  having  no  seas  or  other 
bodies  of  water  upon  it,  and  in  being  more  rugged  and  mountainous.      The  com- 
paratively level  regions  occupy  somewhat  more  than  one-third  of  the  nearer  half 
of  the  moon's  surface.     These  are,  in  general,  the  darker  parts  of  the  disc.     The 
lunar  plains  vary  in  extent  from  40  or  50  miles  to  700  miles  in  diameter. 

316.  The  Mountainous  Formations  of  the  other  parts  of  the 
surface  offer  three  marked  varieties,  viz. : 

(1.)  Insulated  Mountains,  which  rise  from  plains  nearly  level,  and  which  may  be 
supposed  to  present  an  appearance  somewhat  similar  to  Mount  Etna  or  the  Peak 
of  Teneriffe.  The  shadows  of  these  mountains,  in  certain  phases  of  the  moon,  are 
as  distinctly  perceived  as  the  shadow  of  an  upright  staff  when  placed  opposite  to 
the  sun.*'  The  perpendicular  altitudes  of  some  of  them,  as  determined  from  the 
lengths  of  their  shadows,  are  between  four  and  five  miles.  Insulated  mountains 
frequently  occur  in  the  centres  of  circular  plains.  They  are  then  called  Central 
Mountains. 

(2.)  Ranges  of  Mountains,  extending  in  length  two  or  three  hundred  miles.  These 
ranges  bear  a  distinct  resemblance  to  our  Alps,  Apennines,  and  Andes ;  but  they 
are  much  less  in  extent,  and  do  not  form  a  very  prominent  feature  of  the  lunar 

*  Dick's  Celestial  Scenery,  p.  256. 


DESCRIPTION  OF  THE   MOON7S  SURFACE.  189 

surface.  Some  of  them  appear  very  rugged  and  precipitous,  and  the  highest 
ranges  are,  in  some  places,  above  four  miles  in  perpendicular  altitude.  In  some 
instances  they  run  nearly  in  a  straight  line  from  northeast  to  southwest,  as  in  the 
range  called  the  Apennines;  in  other  cases  they  assume  the  form  of  a  semicircle 
or  a  crescent.* 

(3.)  Circular  Formations.  The  general  prevalence  of  this  remarkable  class  of 
mountainous  formations  is  the  great  characteristic  feature  of  the  topography  of  the 
moon's  surface.  It  is  subdivided  by  late  selenographists  into  three  orders,  viz. : 
Waited  Plains,  whose  diameter  varies  from  one  hundred  and  twenty  to  forty  or 
fifty  miles ,  Ring  Mountains,  the  diameter  of  which  descends  to  ten  miles ;  and 
Graters,  which  are  still  smaller.  The  term  crater  is  sometimes  extended  to  all  the 
varieties  of  circular  formations.  They  are  also  sometimes  called  Caverns,  because 
their  enclosed  plains  or  bottoms  are  sunk  considerably  below  the  general  level  of 
the  moon's  surface. 

The  different  orders  of  the  circular  formations  differ  essentially  from  each  other 
only  in  size.  The  principal  features  of  their  constitution  are,  for  the  most  part, 
the  same,  and  they  present  similar  varieties.  Sometimes  terraces  are  seen  going 
round  the  whole  ring.  At  other  times  ranges  of  concentric  mountains  encircle  the 
inner  foot  of  the  wall,  leaving  intermediate  valleys.  Again,  we  have  a  few  ridges 
of  low  mountains  stretching  through  the  circle  contained  by  the  wall,  but  oftener 
isolated  conical  peaks  start  up,  and  very  frequently  small  craters  having  on  an  in- 
ferior scale  every  attribute  of  the  large  one.f  The  smaller  craters,  however,  offer 
some  characteristic  peculiarities.  Most  of  them  are  without  a  flat  bottom,  and 
have  the  appearance  of  a  hollow  inverted  cone  with  the  sides  tapering  towards 
the  centre.  Some  have  no  perceptible  outer  edge,  their  margin  being  on  a  level 
with  the  surrounding  regions :  these  are  called  Pits. 

The  bounding  ridge  of  the  lunar  craters  or  caverns  is  much  more  precipitous 
within  than  without ;  and  the  internal  depth  of  the  crater  is  always  much  lower 
than  the  general  surface  of  the  moon.  The  depth  varies  from  one-third  of  a  mile 
to  three  miles  and  a  half. 

These  curious  circular  formations  oocur  at  almost  every  part  of  the  surface,  but 
are  most  abundant  in  the  southwestern  regions.  It  is  the  strong  reflection  of  their 
mountainous  ridges  which  gives  to  that  part  of  the  moon's  surface  its  superior 
lustre.  The  smaller  craters  occupy  nearly  two-fifths  of  the  moon's  visible  surface. 

*  Dick's  Celestial  Scenery,  p.  257. 

f  Nichol's  Phenomena  of  the  Solar  System,  p.  16T. 


190 


ECLIPSES  OF  THE  SUN  AND  MOON. 


CHAPTER  XVI. 

ECLIPSES  OF  THE  SUN  AND  MOON. — OCCULT ATIONS  OF  THE 
FIXED  STARS. 

317.  AN  eclipse  of  a  heavenly  body  is  a  deprivation  of  its  light, 

occasioned  by  the  interposition  of 
some  opake  body  between  it  and 
the  eye,  or  between  it  and  the  sun. 
Eclipses  are  divided,  with  respect 
to  the  objects  eclipsed,  into  eclipses 
of  the  sun,  of  the  moon,  and  of  the 
satellites ;  and,  with  respect  to  cir- 
cumstances, into  total,  partial,  an- 
nular, and  central.  A  total  eclipse 
is  one  in  which  the  whole  disc  of 
the  luminary  is  darkened;  a  par- 
tial one  is  when  only  a  part  of  the 
disc  is  darkened.  In  an  annular 
eclipse  the  whole  is  darkened,  ex- 
cept a  ring  or  annulus,  which 
appears  round  the  dark  part  like 
an  illuminated  border  ;  the  defini- 
tion of  a  central  eclipse  will  be 
given  in  another  place. 

ECLIPSES  OF  THE  MOOK 

318.  An  eclipse  of  the  moon 
is  occasioned  by  an  interposition 
of  the  body  of  the  earth  directly 
between  the  sun  and  moon,  and 
thus  intercepting  the  light  of  the 
sun  ;  or  the  moon  is  eclipsed  when 
it  passes  through  part  of  the  sha- 
dow of  the  earth,  as  projected  from 
the  sun.  Hence  it  is  obvious  that 
lunar  eclipses  can  happen  only  at 
FIG.  77.  the  time  of  full  moon,  for  it  is 

then  only  that  the  earth  can  be 
between  the  moon  and  the  sun. 

Since  the  sun  is  much  larger  than 


319.  Earth's  Shadow. 


CIRCUMSTANCES   UNDER  WHICH  AN  ECLIPSE   OCCURS.        191 

• 

the  earth,  the  shadow  of  the  earth  must  have  the  form  of  a  cone, 
the  length  of  which  will  depend  on  the  relative  magnitudes 
of  the  two  bodies  and  their  distance  from  each  other.  Let  the 
circles  AGB,  agb  (Fig.  77),  be  sections  of  the  sun  and  earth  by  a 
plane  passing  through  their  centres  S  and  E;  Aa,  Bb.  tangents 
to  these  circles  on  the  same  side,  and  Ac?,  Be,  tangents  on  differ- 
ent sides.  The  triangular  space  aCb  will  be  a  section  of  the 
earth's  shadow  or  Umbra,  as  it  is  sometimes  called.  The  line 
EC  is  called  the  Axis  of-  the  Shadow.  If  we  suppose  the  line  cp 
to  revolve  about  EC,  and  form  the  surface  of  "the  frustum  of  a 
cone,  of  which  pcdq  is  a  section,  the  space  included  within  that 
surface  and  exterior  to  the  umbra,  is  called  the  Penumbra.  It 
is  plain  that  points  situated  within  the  umbra  will  receive  no 
light  from  the  sun;  and  that  points  situated  within  the  penum- 
bra will  receive  light  from  a  portion  of  the  sun's  disc,  and  from 
a  greater  portion  the  more  distant  thev  are  from  the  umbra. 

;32O.  To  fiBBd  the  Length  of  the  Earth's  Shadow. 
Let  L  =  the  length  of  the  shadow  ;  E  =  the  radius  of  the  earth  ; 
$  =  the  sun's  apparent  serni-diameter,  and  p  =.  sun's  parallax. 
The  right-angled  triangle  Ec*C  (Fig.  77)  gives 

EC  = 


sin  ECa 
Ea  =  E  ;  and  EC#  =  SEA  —  E  AC  =  §  —  p  ;  whence, 


As  the  angle  (S  —  p)  is  only  about  16',  it  will  differ  but  little 
from  its  sine,  and  therefore, 

L  =  Rj-l_  (nearly); 

or,  if  S  and  p  be  expressed  in  seconds, 


d  —  p 

The  shadow  will  obviously  be  the  shortest  when  the  sun  is 
nearest  to  the  earth.  We  then  have  $  =  16'  18",  and  p  = 
9",  which  gives  L  =  213  K.  The  greatest  distance  of  the  moon 
is  65R.  It  appears,  then,  that  the  earth's  shadow  always  extends 
to  more  than  three  times  the  distance  of  the  moon. 

321.  Circumstances  under  which  an  Eclipse  occurs. 
Let  kMh  be  a  circular  arc,  described  about  E  the  centre  of  the 
earth,  and  with  a  radius  equal  to  the  distance  between  the  cen- 
tres of  the  earth  and  moon  at  the  time  of  opposition.  The  angle 
MEra,  the  apparent  semi-diameter  of  a  section  of  the  earth's 
shadow,  made  at  the  distance  of  the  moon's  centre,  is  called  the 
Semi-diameter  of  the  Earth's  Shadow.  And  the  angle  ME^,  the 
apparent  semi-diameter  of  a  section  of  the  penumbra,  at  the  same 
distance,  is  called  the  Semi-diameter  of  the  Penumbra. 


192 


ECLIPSES  OF  THE   SUN  AND  MOON. 


"Were  the  plane  of  the  moon's  orbit  coincident  with  the  plane 
of  the  ecliptic,  there  would  be  a  lunar  eclipse  at  every  full 
moon  ;  but,  as  it  is  inclined  to  it,  an  eclipse  can  happen  only 

when  the  full  moon  takes  place  either 
in  one  of  the  nodes  of  the  moon's 
orbit,  or  so  near  it  that  the  moon's 
latitude  does  not  exceed  the  sum  of 
the  apparent  semi-diameters  of  the 
moon  and  of  the  earth's  shadow. 
This  will  be  better  understood  on 
referring  to  Fig.  78,  in  which  N'C 
represents  a  portion  of  the  ecliptic, 
and  N'M  a  portion  of  the  moon's 
orbit,  N'  the  descending  node,  E 
the  earth,  ES,  ES',  ES",  three  dif- 
ferent directions  of  the  sun,  s,  s',  s", 
sections  of  the  earth's  shadow  in  the 
three  several  positions  correspond- 
ing to  these  directions  of  the  sun, 
and  ra,  ra',  ra",  the  moon  in  opposi- 
tion. It  will  be  seen  that  the  moon 
will  not  pass  into  the  earth's  shadow 
unless  at  the  time  of  opposition  it  is 

nearer  to  the  node  than  the  point  m',  where  the  latitude  m's'  is 
equal  to  the  sum  of  the  semi-diameters  of  the  moon  and  shadow. 
0322.  Calculation  of  Semi-diameter  of  Shadow.  To 
determine  the  distance  from  the  node,  beyond  which  there  can 
be  no  eclipse,  we  must  ascertain  the  semi-diameter  of  the  earth's 
shadow.  Let  this  be  denoted  by  A,  arid  let  P  =  the  moon's 
parallax. 

MEra  =  Eraa  —  ECra  (Fig.  77)  ; 
but  Eraa  =  P  and  ECm  =  $—p  (320)  ;  therefore, 

—  5....  (56). 


FIG.  78. 


The  semi-diameter  of  the  shadow  is  the  least  when  the  moon 
is  at  its  greatest  and  the  sun  is  at  its  least  distance,  or  when  P 
has  its  minimum  and  §  its  maximum  value.  In  these  positions 
of  the  moon  and  sun,  P  =  52'  40",  5  =  16'  18",  and  p  =  9". 
Substituting,  we  obtain  for  the  least  semi-diameter  of  the  earth's 
shadow  36'  31",  and  for  its  least  diameter  1°  13'  2".  The  great- 
est apparent  diameter  of  the  moon  is  33'  32".  Whence  it  ap- 
pears that  the  diameter  of  the  earth's  shadow  is  always  more  than 
twice  the  diameter  of  the  moon. 

The  means  of  the  greatest  and  least  values  of  P  and  8  are,  re- 
spectively, 57'  11"  and  16'  2";  which  gives  for  the  mean  semi- 
diameter  of  the  earth's  shadow,  41'  18". 

323.  r  ii  nar  Ecliptic  I.  i  mi  is.  If  to  P  +  p  —  5,  the  semi- 
diameter  of  the  earth's  shadow,  we  add  «?,  the  semi-diameter  of 


PARTICULAR  FACTS.  193 

the  moon,  the  sum  P  +p  +  d  —  $  will  give  the  greatest  latitude 
of  the  moon  in  opposition,  at  which  an  eclipse  can  happen.  It 
is  easy  for  a  given  value  of  P  +  p  •+•  d  —  <$,  and  a  given  inclina- 
tion of  the  moon's  orbit,  to  determine  within  what  distance  from  the 
node  the  moon  must  be  in  order  that  an  eclipse  may  take  place.  By 
taking  the  least  and  greatest  inclinations  of  the  orbit,  the  greatest 
and  least  values  of  F  +  p >  +  d — §,  arid  also  taking  into  view 
the  inequalities  in  the  motions  of  the  sun  and  moon,  it  has  been 
found,  that  when  at  the  time  of  mean  full  moon  the  difference 
of  the  mean  longitudes  of  the  moon  and  node  exceeds  13°  21', 
there  cannot  be  an  eclipse  ;  but  when  this  difference  is  less  than 
7°  47'  there  must  be  one.  Between  7°  47'  and  13°  21'  the  hap- 
pening of  the  eclipse  is  doubtful.  These  numbers  are  called  the 
Lunar  Ecliptic  Limits. 

To  determine,  at  what  full  moons  in  the  course  of  any  one  year 
there  will  be  an  eclipse,  find  the  time  of  each  mean  full  moon  (301) ; 
and  for  each  of  the  times  obtained  find  the  mean  longitude  of  the 
sun,  and  also  of  the  moon's  node,  and  compare  the  difference  of 
these  with  the  lunar  ecliptic  limits.  Should,  however,  the  differ- 
ence in  any  instance  fall  between  the  two  limits,  farther  calculation 
will  be  necessary. 

This  problem  may  be  solved  more  expeditiously  by  means  of 
tables  of  the  sun's  mean  motion  with  respect  to  the  moon's  node. 
(See  Prob.  XXVIII.) 

324.  Central  Eclipse.     The  magnitude  and  duration  of  an 
eclipse  depend  upon  the  proximity  of  the  moon  to  the  node  at  the 
time  of  opposition.     In  order  that  the  centre  of  the  moon  may 
be  on  the  same  right  line  with  the  centres  of  the  sun  and  earth, 
or,  in  technical  language,  that  a  central  eclipse  may  happen,  the 
opposition  must  take  place  precisely  in  the  node.     A  strictly 
central  eclipse,  therefore,  seldom,  if  ever,  occurs.     As  the  mean 
semi-diameter  of  the  earth's  shadow  is  41'  18",  the  mean  semi- 
diameter  of  the  moon  15'  35",  and  the  mean  hourly  motion  of 
the  moon  with  respect  to  the  sun  30'  29",  the  mean  duration  of 
a  central  eclipse  would  be  about  3f  h. 

325.  Particular  Facts.     Since  the  moon  moves  from  west 
to  east,  an  eclipse  of  the  moon  must  commence  on  the  eastern 
limb,  and  end  on  the  western. 

In  the  preceding  investigations,  we  have  supposed  the  cone 
of  the  earth's  shadow  to  be  formed  by  lines  drawn  from  the  edge 
of  the  sun,  and  touching  the  earth's  surface.  This,  probably,  is 
not  the  exact  case  of  nature  ;  for  the  duration  of  the  eclipse,  and 
thus  the  apparent  diameter  of  the  earth's  shadow,  is  found  by 
observation  to  be  somewhat  greater  than  would  result  from  this 
supposition.  This  circumstance  is  accounted  for  by  supposing 
those  solar  rays  that,  from  their  direction,  would  glance  by  and 
raze  the  earth's  surface,  to  be  stopped  and  absorbed  by  the  lower 
strata  of  the  atmosphere.  In  such  a  case  the  conical  boundary 

13 


194  ECLIPSES  OF  THE   SUN  AND  MOON. 

of  the  earth's  shadow  would  be  formed  by  certain  rays  exterior 
to  the  former,  and  would  be  larger. 

The  moon  in  approaching  and  receding  from  the  earth's  total 
shadow,  or  umbra,  passes  through  the  penumbra,  and  thus  its 
light,  instead  of  being  extinguished  and  recovered  suddenly,  ex- 
periences at  the  beginning  of  the  eclipse  a  gradual  diminution, 
and  at  the  end  a  gradual  increase.  On  this  account  the  times  of 
the  beginning  and  end  of  the  eclipse  cannot  be  noted  with  pre- 
cision, and  in  consequence  astronomers  differ  as  to  the  amount 
of  the  increase  in  the  size  of  the  earth's  shadow  from  the  cause 
above  mentioned.  It  is  the  practice,  however,  in  computing  an 
eclipse  of  the  moon,  to  increase  the  semi-diameter  of  the  shadow 
by  a  ^o  part ;  or,  which  amounts  to  the  same,  to  add  as  many 
seconds  as  the  semi-diameter  contains  minutes 

It  is  remarked  in  total  eclipses  of  the  moon,  that  the  moon 
is  not  wholly  invisible,  but  appears  with  a  dull  reddish  light. 
This  phenomenon  is  doubtless  another  effect  of  the  earth's 
atmosphere,  though  of  a  totally  different  nature  from  the  preced- 
ing. Certain  of  the  sun's  rays,  instead  of  being  stopped  and 
absorbed,  are  bent  from  their  rectilinear  course  by  the  refracting 
power  of  the  atmosphere,  so  as  to  form  a  cone  of  faint  light,  in- 
terior to  that  cone  which  has  been  mathematically  described  as 
the  earth's  shadow,  which  falling  upon  the  moon  renders  it 
visible. 

As  an  eclipse  of  the  moon  is  occasioned  by  a  real  loss  of  its 
light,  it  must  begin  and  end  at  the  same  instant,  and  present 
precisely  the  same  appearance  to  every  spectator  who  sees  the 
moon  above  his  horizon  during  the  eclipse.  It  will  be  shown 
that  the  case  is  different  with  eclipses  of  the  sun. 


CALCULATION   OF  AN  ECLIPSE  OF  THE   MOON. 

326.  The  apparent  distance  of  the  centre  of  the  moon  from 
the  axis  of  the  earth's  shadow,  and  the  arcs  passed  over  by  the 
centre  of  the  moon  and  the  axis  of  the  shadow  during  an  eclipse 
of  the  moon,  being  necessarily  small,  they  may,  without  material 
error,  be  considered  as  right  lines.  We  may  also  consider  the 
apparent  motion  of  the  sun  in  longitude,  and  the  motions  of  the 
moon  in  longitude  and  latitude,  as  uniform  during  the  eclipse. 
These  suppositions  being  made,  the  calculation  of  the  circum- 
stances of  an  eclipse  of  the  moon  is  very  simple. 

i*27.  Relative  Orbit.  Let  NF  (Fig.  79)  be  a  part  of  the 
ecliptic,  N  the  moon's  ascending  node,  NL  a  part  of  the  moon's 
orbit,  C  the  centre  of  a  section  of  the  earth's  shadow  at  the  moon, 
CK  perpendicular  to  NF  a  circle  of  latitude,  and  C'  the  centre 
of  the  moon  at  the  instant  of  opposition  :  then  CC',  which  is  the 
latitude  of  the  moon  in  opposition,  is  the  distance  of  the  centres 


CALCULATION  OF  AN  ECLIPSE   OF  THE   MOON. 


195 


of  the  shadow  and  moon  at  that  time.  The  moon  and  shadow 
both  have  a  motion,  and  in  the  same  direction,  as  from  N 
towards  F  and  L.  It  is  the  practice,  however,  to  regard  the 
shadow  as  stationary,  and  to  attribute  to  the  moon  a  motion  equal 


to  the  relative  motion  of  the  moon  and  shadow.  The  orbit  that 
would  be  described  by  the  moon's  centre  if  it  had  such  a  motion, 
is  called  the  Relative  Orbit  of  the  moon.  Inasmuch  as  the  cir- 
cumstances of  the  eclipse  depend  altogether  upon  the  relative 
motion  of  the  moon  and  shadow,  this  mode  of  proceeding  is  obvi- 
ously allowable. 

As  the  shadow  has  no  motion  in  latitude,  the  relative  motion  of 
the  moon  and  shadow  in  latitude  will  be  equal  to  the  moon's  actual 
motion  in  latitude :  and  since  the  centre  of  the  earth's  shadow 
moves  in  the  plane  of  the  ecliptic  at  the  same  rate  as  the  sun,  the 
relative  motion  of  the  moon  and  shadow  in  longitude  will  be  equal 
to  the  difference  between  the  motions  of  the  sun  and  moon  in 
longitude.  We  obtain,  therefore,  the  relative  position  of  the 
centres  of  the  moon  and  shadow  at  any  interval  t,  following  oppo- 
sition, by  laying  off  Cm  equal  to  the  difference  of  the  motions  of 
the  sun  and  moon  in  longitude  in  this  interval,  through  m  draw- 
ing raM  perpendicular  to  NF,  and  cutting  off  raM  equal  to  the 
latitude  at  opposition  plus  the  motion  in  latitude  in  the  interval 
t :  M  will  be  the  position  of  the  moon's  centre  in  the  relative 
orbit,  the  centre  of  the  shadow  being  supposed  to  be  stationary 
at  C.  As  the  motion  of  the  sun  in  longitude,  and  of  the  moon 
in  longitude  and  latitude,  are  considered  uniform,  the  ratio  of  C'm' 
(—  Cm,  the  difference  between  the  motions  of  the  sun  and  moon 
in  longitude)  to  Mm'  the  moon's  motion  in  latitude,  is  the  same, 
whatever  may  be  the  length  of  the  interval  considered.  It  fol- 
lows, therefore,  that  the  relative  orbit  of  the  moon  N'C'M  is  a 
right  line. 

The  relative  orbit  passes  through  C',  the  place  of  the  moon's  centre  at  opposition: 
its  position  will  therefore  be  known,  if  its  inclination  to  the  ecliptic  be  found. 
Now  we  have 


tan  inclina,  =  ™  = 


moon's  motion  in  latitude 


C'm'      moon's  mot.  in  long.  —  sun's  mot.  in  long. 


19(3 


ECLIPSES  OF  THE  SUN   AND  MOON. 


*  32§.  R  equisite  Data.   The  following  data  are  requisite  in  the  calculation 
of  the  circumstances  of  a  lunar  eclipse : 

T  =  time  of  opposition. 

M  =  moon's  hourly  motion  in  longitude. 

n  =  moon's  hourly,  motion  in  latitude. 

m  =  sun's  hourly  motion  in  longitude. 

X  =  moon's  latitude  at  opposition. 

d  =  moon's  semi-diameter. 

3  =  sun's  semi-diameter. 

P  =  moon's  horizontal  parallax. 

p  =  sun's  horizontal  parallax. 

*  =  semi-diameter  of  the  earth's  shadow. 

I  =  inclination  of  relative  orbit. 

h  =  moon's  hourly  motion  on  relative  orbit. 

T,  M,  n,  m,  X,  d,  <T,  P,  and  p,  are  derived  from  Tables  of  the  sun  and  moon.    (See 
Problems  IX  and  XIY.) 

The  quantities  s,  I,  and  h,  may  be  determined  from  these : 

5  =  P  +  p  —  6  +  -^o- (P  +  p  —  J)  (322  and  325). . .  .(51?); 


-_ 

M  —  m 


The  triangle  C'Mw'  gives 


C'M  = 


__,  or,  h  =    -^. . .  .(59). 
cos  MC'w'  cos  I 


v  8559.  Process  of  Calculation.  The  above  quantities  being  supposed 
to  be  known,  let  N'CF  (Fig.  80)  represent  the  ecliptic,  and  G  the  stationary  centre  of 
the  earth's  shadow.  Let  CO'  =  X,  and  let  N'C'L'  represent  the  relative  orbit  of  the 


FiG.  80. 


moon.  "We  here  suppose  the  moon  to  be  north  of  the  ecliptic  at  the  time  of  oppo- 
sition and  near  its  ascending  node ;  when  it  is  south  of  the  ecliptic  X  is  to  be  laid 
off  below  N'CF,  and  when  it  is  approaching  either  node,  the  relative  orbit  is  in- 
clined to  the  right.  Let  the  circle  KFK'R,  described  about  the  centre  0,  repre- 
sent the  section  of  the  earth's  shadow  at  the  moon;  and  let/,  /',  and  g,  g',  be  the 
respective  places  of  the  moon's  centre,  at  the  beginning  and  end  of  the  eclipse,  and 
at  the  beginning  and  end  of  the  total  eclipse.  C/=  C/'  =  s  +  d,  and  00  =  Gg'  = 
s  —  d.  Draw  CM  perpendicular  to  N'C'L',  and  M  will  represent  the  place  of  the 
moon's  centre  when  nearest  the  centre  of  the  shadow:  it  will  also  be  its  place  at 
the  middle  of  the  eclipse;  for  since  Of=  C/',  and  CM  is  perpendicular  to  N'C'/' 
M/  =  M/'. 

Middle  of  the  eclipse.  The  tune  of  opposition  being  known,  that  of  the  middle 
of  the  eclipse  will  become  known  when  we  have  found  the  interval  (x)  employed 
by  the  moon  in  passing  from  M  to  C'.  Now 


CALCULATION  OF  AN  ECLIPSE   OF  THE  MOON.  197 

MO' 

(expressed  in  parts  of  an  hour)  x  =  -  ; 

h 

and  in  the  right-angled  triangle  CC  M  we  have  CO'  =  >,  and  <  C'CM  =  <  C'N'C 
=  I,  and  therefore  MG'  =  A  sin  I  ;  whence,  by  substitution, 

A  sin  I       A  sin  I  X  sin  I  cos  I 


U-m 
cos  I 

or  (expressed  in  seconds),  x  =  3600fl-  °?!J.X  sinl.  .(60). 

Al  ^—  -  7/1 

Hence,  if  M  =  time  of  middle,  we  have 


M  —  m 

It  is  obvious  that  the  upper  sign  is  to  be  used  when  the  latitude  is  increasing, 
and  the  lower  sign  when  it  is  decreasing. 

The  distance  of  the  centre  of  the  moon  from  the  centre  of  the  shadow  at  the 
middle  of  the  eclipse, 

=  CM  =  CO'  cos  C'CM  =  X  cos  I.  .  .  .(62). 

Beginning  and  end  of  the  eclipse.  Let  any  point  I  of  the  relative  orbit  be  the 
place  of  the  moon's  centre  at  the  time  of  any  given  phase  of  the  eclipse.  Let  t  = 
the  interval  of  time  between  the  given  phase  and  the  middle  ;  and  k  =  C/,  the  dis- 
tance between  the  centres  of  the  moon  and  shadow.  In  the  interval  t  the  moon's 
centre  will  pass  over  the  distance  MZ  ;  hence 


M—  m 


but,  MZ  =  \f  C?  —  MC3  =  V  *?  —  >?  cos3 1  (equa.  62), 

and  therefore  t  =     5-L       #  __  j  C08»  l  ; 


or  (in  seconds),       t  =     M_^      |/(&  +  x  cos  I)  (k—X  cos  I). . .  .(63). 

Let  T'  denote  the  tune  of  the  supposed  phase  of  the  eclipse,  and  M  the  time  of 
the  middle ;  and  we  shall  have 

T'  =  M  +  t,  or,  T'  =  M  —  t, 

according  as  the  phase  follows  or  precedes  the  middle. 
Now,  at  the  beginning  and  end  of  the  eclipse,  we  have, 

k  =  CforCf  =  8  +  d: 
substituting  in  equation  (63)  we  obtain 

.-  _  3600s.  cos  I 

\r m —   V  (s  ~r*  d  -f  A  cos  I)  (s  -{-  d  —  A  cos  I). . .  .(64). 


t'  being  found,  the  time  of  the  beginning  (B),  and  the  time  of  the  end  (E),  result 
from  the  equations 


B  =  M—  rf  E  = 

Beginning  and  end  of  the  total  eclipse.  At  the  beginning  and  end  of  the  total 
eclipse,  k  =  Cg  =  Cg'  =  s  —  d  ;  whence,  by  equation  (63), 

,„  _  3600s.  cos  I     ,  _____         ,„.,, 
M_ffl—  \/  (s  —  d  +  A  cos  I)  (s  —  d—  A  cos  I)-  •  •  -(65): 

and,  denoting  the  time  of  the  beginning  by  B'  and  the  time  of  the  end  by  E',  we 
have  B'=M—  «",  E'=M  +  r. 

Quantity  of  the  eclipse.  In  a  partial  eclipse  of  the  moon  the  magnitude  or  quan- 
tity of  the  ech'pse  is  measured  by  the  relative  portion  of  that  diameter  of  the  moon, 
which,  if  produced,  would  pass  through  the  centre  of  the  earth's  shadow,  that  is 
involved  in  the  shadow.  The  whole  diameter  is  divided  into  twelve  equal  parts, 
called  Digits,  and  the  quantity  is  expressed  by  the  number  of  digits  and  fraction? 


198 


ECLIPSES  OF  THE  SUN  AND  MOON. 


of  a  digit  in  the  part  immersed.  When  the  moon  passes  entirely  within  the  sha- 
dow, as  in  a  total  eclipse,  the  quantity  of  the  eclipse  is  expressed  by  the  number  of 
digits  contained  in  the  part  of  the  same  diameter  prolonged  outward,  which  is 
comprised  between  the  edge  of  the  shadow  and  the  inner  edge  of  the  moon. 
Thus  the  number  of  digits  contained  in  SN  (Fig.  80)  expresses  the  quantity  of  the 
eclipse  represented  in  the  figure.  Hence,  if  Q  =  the  quantity  of  the  eclipse,  we 
shall  have 

_  NS    _  12NS  _  12(NM  +  MS)  _  12  (NM  +  CS  —  CM)  __ 
NV  "  NV  NV 

12  (d  +  s  —  A  cos  I) 


or, 


Q  = 


2d~~ 

—  XcosI) 


If  X  cos  I  exceeds  (s  +  d)  there  will  be  no  eclipse.  If  it  is  intermediate  between 
(s  +  d)  and  (5  —  d)  there  will  be  a  partial  eclipse  ;  and  if  it  is  less  than  (s  —  d) 
the  eclipse  will  be  total 


CONSTRUCTION  OF  AN  ECLIPSE  OF  THE  MOON. 

33O.  The  times  of  the  different  phases  of  an  eclipse  of  the 
moon  may  easily  be  determined  by  a  geometrical  construction, 
within  a  minute  or  two  of  the  truth.  Draw  a  right  line  N'F 
(Fig.  81)  to  represent  the  ecliptic ;  and  assume  upon  it  any  point 


C,  for  the  position  of  the  centre  of  the  earth's  shadow,  at  the 
time  of  opposition.  Then,  having  fixed  upon  a  scale  of  equal 
parts,  lay  off  CE,  =  M  —  m,  the  difference  of  the  hourly  motions 
of  the  sun  and  moon  in  longitude  ;  and  draw  the  perpendiculars 
CO'  =  ^  the  moon's  latitude  in  opposition,  and  KI/  =  ^  ±  n. 
the  moon's  latitude  an  hour  after  opposition.  The  right  line 
C'L',  drawn  through  C'  and  I/,  will  represent  the  moon's  relative 
orbit.  It  should  be  observed,  that  if  the  latitudes  are  south  they 
must  be  laid  off  below  N'F,  and  that  N'C'L'  will  be  inclined  to 
the  right  when  the  latitude  is  decreasing.  With  a  radius  CE  = 


ECLIPSES  OF  THE  SUN".  199 

s  (equation  56)  describe  the  circle  EKFK',  which  will  represent 
the  section  of  the  earth's  shadow.  With  a  radius  =  s  +  d,  and 
another  radius  =  s  — c?,  describe  about  the  centre  C  arcs  inter- 
secting N'l/  in//',  and  g,  g' ;  /  and/7  will  be  the  places  of  the 
moon's  centre  at  the  beginning  and  end  of  the  eclipse,  and  g  and 
g'  the  places  at  the  beginning  and  end  of  the  total  eclipse.  From 
the  point  C  let  fall  upon  N'C'L/  the  perpendicular  CM  ;  and  M 
will  be  the  place  of  the  moon's  centre  at  the  middle  of  the  eclipse. 
To  render  the  construction  explicit,  let  us  suppose  the  time  of 
opposition  to  be  7h.  23m.  15s.  At  this  time  the  moon's  centre 
will  be  at  C'.  To  find  its  place  at  7h.,  state  the  proportion, 
60m.  :  23m.  los.  : :  moon's  hourly  motion  on  the  relative  orbit :  a 
fourth  term.  This  fourth  term  will  be  the  distance  of  the  moon's 
centre  from  the  point  C'  at  7  o'clock ;  and  if  it  be  taken  in  the 
dividers  and  laid  off  on  the  relative  orbit  from  C'  backward  to 
the  point  7,  it  will  give  the  moon's  place  at  that  hour.  This 
being  found,  take  in  the  dividers  the  moon's  hourly  motion  on 
the  relative  orbit,  and  lay  it  off  repeatedly,  both  forward  and 
backward,  from  the  point  7,  and  the  points  marked  off,  8,  9,  10, 
6,  5,  will  be  the  moon's  places  at  those  hours  respectively.  Now, 
the  object  being  to  find  the  times  at  which  the  moon's  centre  is 
at  the  points//,  #,  g ',  and  M,  let  the  hour  spaces  thus  found  be 
divided  into  quarters,  and  these  subdivided  into  5-minute  or 
minute  spaces,  and  the  times  answering  to  the  points  of  division 
that  fall  nearest  to  these  points,  will  be  within  a  minute  or  so  of 
the  times  in  question.  For  example,  the  point/  falls  between  9 
and  10,  and  thus  the  end  of  the  eclipse  will  occur  somewhere 
between  9  and  10  o'clock.  To  find  the  number  of  minutes  after 
9  at  which  it  takes  place,  we  have  only  to  divide  the  space  from 
9  to  10  into  four  equal  parts,  or  15-minute  spaces,  subdivide  the 
part  which  contains/  into  three  equal  parts,  or  5-minute  spaces, 
and  again  that  one  of  these  smaller  parts  within  which  /  lies, 
into  five  equal  parts  or  minute  spaces. 


ECLIPSES  OF  THE  SUN. 

331.  1, 11  mi  11011*  Frustum  and  Coue.  An  eclipse  of  the 
sun  is  caused  by  the  interposition  of  the  moon  between  the  sun 
and  earth  ;  whereby  the  whole,  or  part  of  the  sun's  light,  is  pre- 
vented from  falling  upon  certain  parts  of  the  earth's  surface. 

Let  AGB  and  agb  (Fig.  82)  be  sections  of  the  sun  and  earth 
by  a  plane  passing  through  their  centres  S  and  E ;  Aa,  B5,  tan- 
gents to  the  circles  AGB  and  agb  on  the  same  side;  and  Ad,  Be, 
tangents  to  the  same  on  opposite  sides.  The  figure  Aa&B  will 
be  a  section  through  the  axis,  of  a  frustum  of  a  cone  formed  by- 
rays  tangent  to  the  sun  and  earth  on  the  same  side,  and  the  tri- 
angular space  Ycd  will  be  a  section  of  a  cone  formed  by  rays 


200  ECLIPSES   OF  THE  SUN  AND   MOON. 

tangent  on  opposite  sides.  An  eclipse  of  the  sun  will  take  place 
somewhere  upon  the  earth's  surface,  whenever  the  moon  comes 
within  the  frustum  Aa&B,  and  a  total  or  an  annular  eclipse  when- 
ever it  comes  within  the  cone  Fee?. 


FIG.  82. 

332.  Semi-diameters  of  Frustum  and  Cone.  Let  mra'M 
(Fig  82)  be  a  circular  arc  described  about  the  centre  E,  and  with 
a  radius  equal  to  the  distance  between  the  centres  of  the  moon  and 
earth  at  the  time  of  conjunction.     The  angle  raES  is  the  apparent 
semi-diameter  of  a  section  of  the  frustum,  and  m'ES  the  apparent 
semi-diameter  of  a  section  of  the  cone,  at  the  distance  of  the 
moon.     To  find  expressions  for  these  semi-diameters  in  terms  of 
determinate  quantities,  let  the  first  be  denoted  by  A,  and  the 
second  by  A7 ;  and  let  P  =  the  parallax  of  the  moon,  p  =  the 
parallax  of  the  sun,  and  8  =  the  semi-diameter  of  the  sun.     Then 
we  have 

raES  =  A  =  raEA  +  AES  =  Eraa  — EAm  +  AES; 
or,  A  =  P  —  ^ +  «.-.. (67): 

and        ra'ES  =  ra'EB  —  BES  =  Era'c  — EBra'  —  BES ; 
or,  A'  =  P  —  p  —  5.... (68). 

Taking  the  mean  values  of  P,  ^,  and  8  (322),  we  find  for  the 
mean  value  of  A,  1°  13'  3" ;  and  for  the  mean  value  of  A'. 
41'  1". 

333.  Circumstances    of     Moon's     Position    in    Solar 
Eclipses.     As  the  plane  of  the  moon's  orbit  is  not  coincident 
with  the  plane  of  the  ecliptic,  an  eclipse  of  the  sun  can  happen 
only  when  conjunction  or  new  moon  takes  place  in  one  of  the 
nodes  of  the  moon's  orbit,  or  so  near  it  that  the  moon's  latitude 
does  not  exceed  the  sum  of  the  semi-diameters  of  the  moon 
and  luminous  frustum  at  the  moon's  orbit.     This  may  be  illus- 
trated by  means  of  Fig.  78,  already  used  for  a  lunar  eclipse, 
by  supposing  the  sun  to  be  in  the  directions  Es,  Es',  E*",  and 
that  5,  s',  s",  are  sections  of  the  luminous  frustum  corresponding 
to  these  directions  of  the  sun  ;  also  that  ra,  ra',  m" ,  represent  the 
moon  in  the  corresponding  positions  of  conjunction.     Thus,  de- 


NUMBER  OF  ECLIPSES  IS  A  YEAR.  201 

noting  the  moon's  semi-diameter  by  d,  and  the  greatest  latitude 
of  the  moon  in  conjunction,  at  which  an  eclipse  can  take  place, 
by  L,  we  have 

L  =  P  —  p  +3  +  <*....  (69). 

For  a  total  eclipse,  the  greatest  latitude  will  be  equal  to  the  sum 
of  the  semi-diameters  of  the  moon  and  the  luminous  cone.  Hence, 
denoting  it  by  L', 

L'  =  P—  p—  J$  +  d....(70). 

In  order  that  an  annular  eclipse  may  take  place,  the  apparent 
semi-diameter  of  the  moon  must  be  less  than  that  of  the  sun,  and 
the  moon  must  come  at  conjunction  entirely  within  the  luminous 
frustum.  Whence,  if  L"=  the  maximum  latitude  at  which  an 
annular  eclipse  is  possible,  we  have 


In  the  same  manner  as  in  the  case  of  an  eclipse  of  the  moon, 
it  has  been  found  that  when  at  the  time  of  mean  new  moon  the 
difference  between  the  mean  longitude  of  the  sun  or  moon  and 
that  of  the  node,  exceeds  19°  44',  there  cannot  be  an  eclipse  of  the 
sun  ;  but  when  the  difference  is  less  than  13°  33',  there  must  be 
one.  These  numbers  are  called  the  Solar  Ecliptic  Limits. 

334.  Prediction  of  Eclipses  :—  Period.     In  order  to  dis- 
cover at  what  new  moons  in  the  course  of  a  year  an  eclipse  of  the 
sun  will  happen,  with  its  approximate  time,  we  have  only  to  find 
the  mean  longitudes  of  the  sun  and  node  at  each  mean  new  moon 
throughout  the  year  (301),  and  take  the  difference  of  the  longi- 
tudes and  compare  it  with  the  solar  ecliptic  limits.     (For  a  more 
direct  method  of  solving  this  problem,  see  Prob.  XXVIII.) 

Eclipses  both  of  the  sun  and  moon  recur  in  nearly  the  same 
order  and  at  the  same  intervals  at  the  expiration  of  a  period  of 
223  lunations,  or  18  years  of  365  days,  and  15  days;*  which 
for  this  reason  is  called  the  Period  of  the  Eclipses.  For,  the  time 
of  a  revolution  of  the  sun  with  respect  to  the  moon's  node  is 
346.619S51d.,  and  the  time  of  a  synodic  revolution  of  the  moon 
is  29.5305887d.  These  numbers  are  very  nearly  in  the  ratio  of 
223  to  19.  Thus,  in  a  period  of  223  lunations,  the  sun  will  have 
returned  19  times  to  the  same  position  with  respect  to  the  moon's 
node,  and  at  the  expiration  of  the  period  will  be  in  the  same 
position  with  respect  to  the  moon  and  node  as  at  its  commence- 
ment. The  eclipses  which  occur  during  one  such  period  being 
noted,  subsequent  eclipses  are  easily  predicted. 

This  period  was  known  to  the  Chaldeans  and  Egyptians,  by 
whom  it  was  called  Saros. 

335.  Number  of  Eclipses  in  a  Tear.     As  the  solar  eclip- 
tic limits  are  more  extended  than  the  lunar,  eclipses  of  the  sun 
must  occur  more  frequently  than  eclipses  of  the  moon. 

*  More  exactly,  18  years  (of  365  days)  plus  15d.  7h.  42m.  29s. 


202  ECLIPSES  OF  THE  SUN    AND  MOON. 

As  to  the  number  of  eclipses  of  both  luminaries,  there  cannot  be 
fewer  than  two  nor  more  than  seven  in  one  year.  The  most 
usual  number  is  four,  and  it  is  rare  to  have  more  than  six. 
"When  there  are  seven  eclipses  in  a  year,  five  are  of  the  sun  and 
two  of  the  moon ;  and  when  but  two,  both  are  of  the  sun.  The 
reason  is  obvious.  The  sun  passes  by  both  nodes  of  the  moon's 
orbit  but  once  in  a  year,  unless  it  passes  by  one  of  them  in  the 
beginning  of  the  year,  in  which  case  it  will  pass  by  the  same 
again  a  little  before  the  end  of  the  year,  as  it  returns  to  the 
same  node  in  a  period  of  346  days.  Now,  if  the  sun  be  at  a  little 
less  distance  than  19°  44'  from  either  node  at  the  time  of  mean 
new  moon,  he  may  be  eclipsed  (333),  and  at  the  subsequent  op- 
position the  moon  will  be  eclipsed  near  the  other  node,  and  come 
round  to  the  next  conjunction  before  the  sun  is  13°  33'  from  the 
former  node ;  and  when  three  eclipses  happen  about  either  node, 
the  like  number  commonly  happens  about  the  opposite  one ;  as 
the  sun  comes  to  it  in  173  days  afterwards,  and  six  lunations 
contain  only  four  days  more.  Thus  there  may  be  two  eclipses 
of  the  sun  and  one  of  the  moon  about  each  of  the  nodes ;  and  the 
twelfth  lunation  from  the  eclipse  in  the  beginning  of  the  year 
may  give  a  new  moon  before  the  year  is  ended,  which,  in  conse- 
quence of  the  retrogradation  of  the  nodes,  may  be  within  the 
solar  ecliptic  limit ;  and  hence  there  may  be  seven  eclipses  in  a 
year,  five  of  the  sun  and  two  of  the  moon.  But  when  the  moon 
changes  in  either  of  the  nodes,  it  cannot  be  near  enough  to  the 
other  node,  at  the  next  full  moon,  to  be  eclipsed ;  as  in  the  inter- 
val the  sun  will  move  over  an  arc  of  14°  32',  whereas  the  great- 
est lunar  ecliptic  limit  is  but  13°  21',  and  in  six  lunar  months 
afterwards  it  will  change  near  the  other  node.  In  this  case 
there  cannot  be  more  than  two  eclipses  in  a  year,  both  of  which 
will  be  of  the  sun.  If  the  moon  changes  at  the  distance  of  a  few 
degrees  from  either  node,  then  an  eclipse  both  of  the  sun  and 
moon  will  probably  occur  in  the  passage  of  that  node  and  also 
of  the  other. 

Although  solar  eclipses  are  more  frequent  than  lunar,  when 
considered  with  respect  to  the  whole  earth,  yet  at  any  given 
place  more  lunar  than  solar  eclipses  are  seen.  The  reason  of  this 
circumstance  is,  that  an  eclipse  of  the  sun  (unlike  an  eclipse  of 
the  moon)  is  visible  only  over  a  part  of  a  hemisphere  of  the 
earth.  To  show  this,  suppose  two  lines  to  be  drawn  from  the 
centre  of  the  moon  tangent  to  the  earth  at  opposite  points :  they 
will  make  an  angle  with  each  other  equal  to  double  the  moon's 
horizontal  parallax,  or  of  1°  54'.  Therefore,  should  an  observer 
situated  at  one  of  the  points  of  tangency,  refer  the  centre  of  the 
moon  to  the  centre  of  the  sun,  an  observer  at  the  other  would 
see  the  centres  of  these  bodies  distant  from  each  other  an  angle 
of  1°  54',  and  their  nearest  limbs  separated  by  an  arc  of  more 
than  1°.  ^ 


MOONS  SHADOW  CAST  UPON  THE  EARTH. 


203 


Moon's  Shadow  Cast  upon  the  Earth.  Instead  of 
regarding  an  eclipse  of  the  sun  as  produced  by  an  interposition 
ofthe  moon  between  the  sun  and  earth,  as  we  have  hitherto 
considered  it,  we  may  regard  it  as  occasioned  by  the  moou'^s 
shadow  falling  upon  the  earth.  Fig.  83  represents  the  moon's 
shadow,  as  projected  from  the  sun  and  covering  a  portion 
of  the  earth's  surface.  Wherever  the  umbra  falls,  there  is  total 
eclipse ;  and  wherever  the  penum,bra  falls,  a  partial  eclipse. 


FIG.  83. 

In  order  to  discover  the  extent  of  the  portion  of  the  earth's 
surface  over  which  the  eclipse  is  visible  at  any  particular  time, 
we  have  only  to  find  the  breadth  of  the  portion  of  the  earth 
covered  by  the  penumbral  shadow  of  the  moon ;  but  we  will 
first  ascertain  the  length  of  the  moon's  shadow.  As  seen  at  the 
vertex  of  the  moon's  shadow,  the  apparent  diameters  of  the  moon, 
and  sun  are  equal.  Now,  as  seen  at  the  centre  of  the  earth,  they 
are  nearly  equal,  sometimes  the  one  being  a  little  greater  and 
sometimes  the  other.  It  follows,  therefore,  that  the  length  of  the 
moon's  shadow  is  about  equal  to  the  distance  of  the  earth,  being  some- 
times a  little  greater  and  at  other  times  a  little  less. 

When  the  apparent  diameter  of  the  moon  is  the  greater,  the 
shadow  will  extend  beyond  the  earth's  centre;  and  when  the 
apparent  diameter  of  the  sun  is  the  greater,  it  will  fall  short  of  it. 
If  we  increase  the  mean  apparent  diameter  of  the  moon  as  seen 
from  the  earth's  centre,  viz.  31/  7",  by  -g^,  the  ratio  of  the  radius 
of  the  earth  to  the  distance  of  the  moon,  we  shall  have  31'  38" 
for  the  mean  apparent  diameter  of  the  moon  as  seen  from  the 
nearest  point  of  the  earth's  surface.  Comparing  this  with  the 
mean  apparent  diameter  of  the  sun  as  viewed  from  the  same 
point,  which  is  sensibly  the  same  as  at  the  centre  of  the  earth,  or 
32'  3",  we  perceive  that  it  is  less ;  from  which  we  conclude,  that 
when  the  sun  and  moon  are  each  at  their  mean  distance  from  the 
earth,  the  shadow  of  the  moon  does  not  extend  as  far  as  the 
earth's  surface. 


204:  ECLIPSES  OF  THE  SUN  AND  MOON. 

337.  To  fiaid  a  General  Expression  for  the  length  of 
the  Moon's  Shadow.  Let  AGB,  dg'V,  and  agb  (Fig.  84)  be 
sections  of  the  sun,  moon,  and  earth,  by  a  plane  passing  through 


PIG.  84. 

their  centres  S,  M,  and  E,  supposed  to  be  in  the  same  right  line, 
and  Aa',  B6',  tangents  to  the  circles  AGB,  a'y'V :  then  a'Kb' 
will  represent  the  moon's  shadow.  Let  L  =  the  length  of  the 
shadow ;  D  =  the  distance  of  the  moon ;  D'  =  the  distance  of 
the  sun ;  d  =  the  apparent  semi-diameter  of  the  moon  ;  and  i  = 
the  apparent  semi-diameter  of  the  sun.  At  K  the  vertex  of  the 
shadow,  MKa'  the  apparent  semi-diameter  of  the  moon,  will  be 
equal  to  SKA  the  apparent  semi-diameter  of  the  sun  ;  and  as  the 
distance  of  this  point  from  the  centre  of  the  earth,  even  when  it 
is  the  greatest,  is  small  in  comparison  with  the  distance  of  the 
sun  (336),  the  apparent  semi- diameter  of  the  sun  will  always  be 
very  nearly  the  same  to  an  observer  situated  at  K  as  to  one  situ- 
ated at  the  centre  of  the  earth.  Now,  since  the  apparent  semi- 
diameter  of  the  moon  is  inversely  proportional  to  its  distance, 

angle  MKa'  :  d ::  ME  :  MK; 
and  thus,        J  :  d  : :  ME  :  MK  : :  D  :  L  (nearly) : 

whence,  L  =  D^ . . . .  (72). 

0 

If  a  more  accurate  result  be  desired,  we  have  only  to  repeat 
the  calculation,  after  having  diminished  <J  in  the  ratio  of  D'  to 
(D'  +  L  —  D). 

9  338.  To  find  the  Breadth  of  the  Peiiumbral  Shadow 

cast  upon  the  earth,  let  the  lines  Ad',  Be'  (Fig.  84)  be  drawn  tangent  to  the  circles 
AG-B,  a'g'b\  on  opposite  sides,  and  prolonged  to  the  earth.  The  space  hc'd'k  will 
represent  the  penumbra  of  the  moon's  shadow,  and  the  arc  gh  one-half  the  breadth 
of  the  portion  of  the  earth's  surface  covered  by  it.  Let  this  arc  or  the  angle 
gWi  =  S,  and  denote  the  semi-diameter  of  the  sun  and  the  semi-diameter  and 
parallax  of  the  moon  by  the  same  letters  as  in  previous  articles.  The  triangle 
ME&  gives 

angle  ME/i  =  S  =  MAZ  —  AME. 

The  angle  7tME  is  the  moon's  parallax  in  altitude  at  the  station  h,  and  WiZ  is 
its  zenith  distance  at  the  same  station.  Denote  the  former  by  P'  and  the  latter 
byZ.  Thus, 

S  =  Z— P'.. 
The  triangle  ftMS  gives 


LENGTH  AND  BREADTH  OF  MOON'S  SHADOW.  205 

MAS  =  d  +  J;  and  MSft  is  the  sun's  parallax  in  altitude  at  the  station  h  :  let  it  be 
denoted  by  p.     "Wo  have,  then, 

P'  =  d  +  6+p'  =  d+i  (nearly)  ----  (74); 
aud  to  find  Z  we  have  (equa,  9,  p.  63), 

P'  =  P  sin  Z,  or  sin  Z  =  ?!...  .(75). 

P'  and  Z  being  found  by  these  equations,  equa.  (73)  will  then  make  known  the 
value  of  S. 

If  great  accuracy  be  required,  the  calculation  must  be  repeated,  giving  now  to 
p'  in  equa.  (74)  the  value  furnished  by  equa.  (9)  which  expresses  the  relation 
between  the  parallax  in  altitude  of  a  body  and  its  horizontal  parallax,  instead  of 
neglecting  it  as  before;  and  Z  must  be  computed  from  the  following  equation: 

Bin  Z  =  !**:....  (76). 
sin  P 

The  penumbral  shadow  will  obviously  attain  to  its  greatest 
breadth  when  the  sun  is  at  its  least  and  the  moon  is  at  its  great- 
est distance.  The  values  of  c?,  <5,  and  P  under  these  circumstances 
are  respectively  14'  24",  16'  18",  and  52'  50".  Performing  the 
calculations,  we  find  that  the  breadth  of  the  greatest  portion  of  the 
earth's  surface  ever  covered  by  the  penumbral  shadow  is  70°  17',  or 
about  4,850  miles. 

0339.  The  Breadth  of  the  Umbra  may  be  found  in  a  simi- 
lar manner. 

The  arc  gh'  (Fig.  84)  represents  one  half  of  it  :  denote  this  arc  or  the  equal  angle 
gYJi  by  S'. 

MEft'  =  S'  =  Mfc'Z'  —  A'ME  ; 
or,  S'  =  Z  —  F....(77). 


but  MA'S  =  d  —  J,  and  MSft'  =  p',  the  sun's  parallax  in  altitude  at  h'  ;  whence, 

P'  =  d  —  6  +p'  =  d  —  6  (nearly).  .  .  .(78): 
and  we  have,  as  before, 

P'  =  P  sin  Z,  or  sin  Z  =  ?'...  .(79). 

The  greatest  breadth  will  obtain  when  the  sun  is  at  its  greatest 
and  the  moon  is  at  its  least  distance.  We  shall  then  have 

3  =  15'  45",  d=  16'  46",  P  =  61'  32". 

Making  use  of  these  numbers,  we  deduce  for  the  maximum- 
breadth  of  the  portion  of  the  earth's  surface  covered  by  the  moon's 
shadow,  1°  54'  ;  or  130  miles. 

It  should  be  observed  that  the  deductions  of  the  last  two  arti- 
cles answer  to  the  supposition  that  the  moon  is  in  the  node,  and 
that  the  axis  of  the  shadow  and  penumbra  passes  through  the 
centre  of  the  earth.  In  every  other  case,  both  the  shadow  and 
penumbra  will  be  cut  obliquely  by  the  earth's  surface,  and  the 
sections  will  be  ovals,  and  very  nearly  true  ellipses,  the  lengths 
of  which  may  materially  exceed  the  above  determinations. 

34O.  Phase§  of  Eclipse  Different  at  each  Place.  Parallax 
not  only  causes  the  eclipse  to  be  visible  at  some  places  and  invisi- 
ble at  others,  as  shown  in  Art.  335,  but,  by  making  the  distance 


206  ECLIPSES  OF  THE   SUN  AND   MOON. 

between  the  centres  of  the  sun  and  moon  unequal,  renders  the  cir- 
cumstances of  the  eclipse  at  those  places  where  it  is  visible,  differ- 
ent at  each  place.  This  may  also  be  inferred  from  the  circumstance 
that  the  different  places,  covered  at  any  time  by  the  shadow  of 
the  moon,  will  be  differently  situated  within  this  shadow.  It 
will  be  seen,  therefore,  that  an  eclipse  of  the  sun  has  to  be  consid- 
ered in  two  points  of  view :  1st.  With  respect  to  the  whole  earth, 
or  as  a  general  eclipse;  and,  2d.  With  respect  to  a  particular 
place. 

341.  Particular  Facts.  The  following  are  the  principal  facts  relative 
to  eclipses  of  the  sun  that  remain  to  be  noticed  :  1st.  The  duration  of  a  general 
eclipse  of  the  sun  cannot  exceed  about  6  hours.  2d.  A  solar  eclipse  does  not  hap- 
pen at  the  same  time  at  all  places  where  it  is  seen :  as  the  motion  of  the  moon 
toward  the  sun,  and  consequently  of  its  shadow,  is  from  west  to  east,  the  eclipse 
must  begin  earlier  at  the  western  parts  and  later  at  the  eastern.  3d.  The  moon's 
shadow  being  tangent  to  the  earth  at  the  commencement  and  end  of  the  eclipse, 
the  sun  will  be  just  rising  at  the  place  where  the  eclipse  is  first  seen,  and  just  set- 
ting at  the  place  where  it  is  last  seen.  At  intermediate  places,  the  sun  will 
at  the  time  of  the  beginning  and  end  of  the  eclipse  have  various  altitudes.  4th. 
An  eclipse  of  the  sun  begins  on  the  western  side  and  ends  on  the  eastern.  5th. 
When  the  straight  line  passing  through  the  centres  of  the  sun  and  moon  passes 
also  through  the  place  of  the  spectator,  the  eclipse  is  said  to  be  central :  a  central 
eclipse  may  be  either  annular  or  total,  according  as  the  apparent  diameter  of  the 
sun  is  greater  than  that  of  the  moon,  or  the  reverse.  6th.  A  total  eclipse  of  the  sun 
cannot  last  at  any  one  place  more  than  eight  minutes;  and  an  annular  eclipse  moro 
than  twelve  and  a  half  minutes.  7th.  In  most  solar  eclipses  the  moon's  disc  is 
covered  with  a  faint  light,  a  phenomenon  which  is  attributed  to  the  reflection  of 
the  light  from  the  illuminated  part  of  the  earth. 


CALCULATION  OP  AN  ECLIPSE  OF  THE  SUN. 

342.  The  complete  calculation  of  a  solar  eclipse  involves  the 
solution  of  two  distinct  problems,  viz. :  (1),  the  determination  of 
all  the  circumstances  of  the  eclipse  for  the  earth  as  a  whole ;  (2), 
the  determination  of  the  times  of  all  the  phases,  and  the  corre- 
sponding apparent  relative  positions  of  the  moon  and  sun  for  a 
particular  place.     Different  methods  of  solving  these  problems 
Lave  been  devised.     Processes  of  calculation,  comparatively  sim- 
ple and  direct,  are  given  in  the  Appendix. 

OCCULTATIONS. 

343.  An  occultation  is  an  eclipse,  or  deprivation  of  the  light 
of  a  star,  resulting  from  the  interposition  of  the  moon  between 
the  star  and  the  eye  of  the  observer.     At  all  places  on  the  earth 
which  at  a  given  time  have  the  moon  in  the  horizon,  its  apparent 
place  will  differ  from  its  true  place  (78),  by  the  amount  of  the 
horizontal  parallax.     It  follows,  therefore,  that  a  star  will  be 
eclipsed  by  the  moon,  somewhere  upon  the  earth,  in  case  its  true 
distance  from  the  moon's  centre  is  less  than  the  sum  of  the  moon's 
semi-diameter  and  horizontal  parallax. 


OCCULTATTONS.  207 

344.  Limit§  of  Position  of  Stars  Liable  to  Occultation. 

The  greatest  value  of  the  apparent  semi-diameter  of  the  moon  is 
16'  46",  and  that  of  its  horizontal  parallax  is  61'  32".  If  we  add 
the  sum  of  these  quantities  to  5°  20'  6",  the  greatest  possible 
latitude  of  the  moon,  we  obtain  as  the  result  6°  38'  24".  It  is 
then  only  the  stars  which  have  a  latitude,  either  north  or  south, 
less  than  6°  38'  24"  that  can  experience  an  occultation  from  the 
moon. 

In  order  that  any  of  the  stars  situated  within  this  distance 
from  the  ecliptic  may  suffer  occultation  at  some  point  on  the 
earth,  it  is  necessary  that,  at  the  time  of  the  true  conjunction  (144) 
of  the  moon  and  star,  the  actual  difference  of  latitude  of  the 
two  should  not  exceed  the  sum  of  the  actual  apparent  semi-dia- 
meter and  horizontal  parallax  of  the  moon. 


208 


THE  PLANETS. 


CHAPTER  XVII. 

THE  PLANETS,  AND  THE  PHENOMENA  OCCASIONED  BY  THEIR 
MOTIONS  IN  SPACE. 

APPARENT  MOTIONS  OF  THE  PLANETS  WITH  RESPECT  TO 

THE  SUN 

345.  THE  apparent  motion  of  an  inferior  planet  with  reference 
to  the  sun,  is  materially  different  from  that  of  a  superior  planet. 
The  inferior  planets  always  accompany  the  sun,  being  seen  alter- 
nately on  the  east  and  west  side  of  it,  and  never  receding  from 
it  beyond  a  certain  moderate  distance,  while  the  superior  planets 
are  seen,  at  different  times,  at  every  variety  of  angular  distance. 
This  difference  of  apparent  motion  arises  from  the  difference  of 
situation  of  the  orbits  of  an  inferior  and  superior  planet,  with 
respect  to  the  orbit  of  the  earth  ;  the  one  lying  within,  and  the 
other  without  the  earth's  orbit. 


340.  Apparent  Motion   of   an   Inferior  Planet.     Let 

C  AC'B  (Fig.  85)  represent  the  orbit  of  either  one  of  the  inferior 


MOTION  OF  A  SUPERIOR  PLANET.  209 

planets,  Venus  for  example,  and  PKT  the  orbit  of  the  earth — 
which  we  will  suppose  to  be  circles,  and  to  lie  in  the  same  plane, 
— and  let  MLNjcepresent  the  circle  of  intersection  of  tbjs_plane_ 
with  the  sphejyof  the  h^av^reyt"  some  point  of  which  the  planet 
will  be  referred  by  an  observer  on  the  earth.  Suppose,  for  the 
present,  that  the  earth  is  stationary  in  the  position  P,  and  through 
P  draw  the  lines  PA,  PB,  tangent  to  the  orbit  of  Yenus,  and 
prolong  them  until  they  intersect  the  heavens  at  a  and  b.  The 
earth  being  at  P,  Venus  will  be  in  superior  conjunction  at  C, 
and  in  inferior  conjunction  at  C'.  Now,  by  inspecting  the  figure, 
it  will  be  seen  that  in  passing  from  C  to  C'  she  will  be  seen  in 
the  heavens  on  the  east  side  of  the  sun,  and  in  passing  from  C' 
to  C  on  the  west  side  of  the  sun ;  also,  that  in  passing  from  C  to 
A  she  will  recede  from  the  sun  in  the  heavens,  from  A  to  C' 
approach  it,  from  C'  to  B  recede  from  it  again,  and  from  B  to  C 
approach  it  again,  a  and  b  will  be  the  positions  of  the  planet  in 
the  heavens  at  the  times  of  the  greatest  eastern  and  western, 
elongations. 

When  to  the  east  of  the  sun,  Venus  is  seen  in  the  evening,  and 
called  the  Evening  Star;  and  when  to  the  west  of  the  sun,  is  seen 
in  the  morning,  and  called  the  Morning  Star. 

We  have  in  the  foregoing  investigation  supposed  the  earth  to 
be  stationary,  a  supposition  which  is  contrary  to  the  fact;  but  it 
is  plain  that  the  only  effect  of  the  earth's  motion  in  the  case 
under  consideration,  as  it  is  slower  than  that  of  the  planet,  is  to 
cause  the  points  A,  C',  B,  to  advance  in  the  orbit,  without  alter- 
ing the  nature  of  the  apparent  motion  of  the  planet  with  respect 
to  the  sun.  The  orbits  of  the  earth  and  planet  are  also  ellipses 
of  small  eccentricity,  and  are  slightly  inclined  to  each  other, 
instead  of  being  circles  and  lying  in  the  same  plane :  on  this 
account,  as  the  greatest  elongations  will  occur  in  various  parts 
of  the  orbits,  they  will  differ  in  value.  The  greatest  elonga- 
tion of  Venus  varies  from  45°  to  47°  15'.  Its  mean  value  is 
about  46°. 

Owing  to  the  circumstance  of  the  orbit  of  Mercury  being 
within  the  orbit  of  Venus,  the  greatest  elongation  of  this  planet 
is  less  than  that  of  Venus.  It  is  never  so  great  as  30°. 

34T.  Apparent  ?Iotioii  of  a  Superior  Planet.  Suppose 
PKT  (Fig.  85)  to  be  the  orbit  of  a  superior  planet,  and  CAC'B 
that  of  the  earth  ;  and  as  the  velocity  of  the  earth  is  much  greater 
than  that  of  the  planet,  let  us,  for  the  present,  regard  the  planet 
as  stationary  in  the  position  P,  while  the  earth  describes  the  cir- 
cle CACT.  When  the  earth  is  at  C,  the  planet  being  at  P,  is  in 
conjunction  with  the  sun.  When  the  earth  is  at  A,  SAP,  the 
elongation  of  the  planet,  is  90°.  When  it  arrives  at  C',  the  planet 
is  in  opposition,  or  180°  distant  from  the  sun  :  and  when  it  reaches 
B,  the  elongation  is  again  90°.  At  intermediate  points,  the  elon- 
gation will  have  intermediate  values.  If,  now,  we  restore  to  the 

14 


210  THE    PLANETS. 

planet  its  orbital  motion,  we  shall  manifestly  be  conducted  to  the 
same  results  relative  to  the  change  of  elongation,  as  the  only 
effect  of  such  motion  will  be  to  throw  the  points  A,  C',  B,  forward 
in  the  orbit.  It  appears,  then,  that  in  the  course  of  a  synodic 
revolution  a  superior  planet  will  be  seen  at  all  angular  distances 
from  the  sun,  both  on  the  east  and  west  side  of  him.  From  con- 
junction to  opposition,  that  is,  while  the  earth  is  passing  from 
C  to  C',  the  planet  will  be  to  the  right,  or  to  the  west  of  the  sun  ; 
and  will  therefore  be  below  the  horizon  at  sunset,  and  rise  some 
time  in  the  course  of  the  night.  But,  from  opposition  to  conjunc- 
tion, or  while  the  earth  is  moving  from  C'  to  C,  it  will  be  to  the 
east  of  the  sun,  and  therefore  above  the  horizon  at  sunset. 

-,34§.  To  find  the  Length  of  the  Synodic  Revolution 
of  a  Planet.  Let  us  first  take  an  inferior  planet,  Yenus  for 
instance.  Suppose  we  assume,  at  a  given  instant,  the  sun,  Yenus, 
and  the  earth  to  be  in  the  same  right  line  ;  then,  after  any  elapsed 
time  (a  day  for  instance),  Yenus  will  have  described  an  angle  m, 
and  the  earth  an  angle  M  around  the  sun.  Now,  the  value  of  m 
is  greater  than  that  of  M  ;  therefore,  at  the  end  of  a  day,  the 
separation  of  the  planet  from  the  earth  (measuring  the  separation 
by  an  angle  formed  by  two  lines  drawn  from  the  planet  and  the 
earth  to  the  sun),  will  be  m  —  M  ;  at  the  end  of  two  days  (the 
mean  daily  motions  continuing  the  same)  the  angle  of  separation 
wil  be  2  (m  —  M)  ;  at  the  end  of  three  days,  8  (m  —  M)  ;  at  the 
end  of  s  days,  s  (m  —  M).  When  the  angle  of  separation 
amounts  to  360°,  that  is,  when  s  (m  —  M)  =  360°,  the  sun, 
planet,  and  earth  must  be  again  in  the  same  right  line,  and  in 
that  case 


In  which  expression  s  denotes  the  mean  duration  of  a  synodic 
revolution,  m  and  M  being  taken  to  denote  the  mean  daily 
motions. 

We  may  obtain  from  equation  (80)  another  equation,  in  which 
the  synodic  revolution  is  expressed  in  terms  of  the  sidereal  peri- 
ods of  the  earth  and  planet. 

Let  P  and^  denote  the  sidereal  periods  in  question,  then,  since 

Id.  :  M°  :  :  P  :  360°, 
1     :  m    :  :  p  :  360  ; 

and  m  =  8^!;  substituting, 


_  _=_-       .(81). 

-4)  p~f 

Equations  (80),  (81),  although  investigated  for  an  inferior 


STATIONS  AND   RETROGRADATIONS  OF  THE   PLANETS.     211 

planet,  will  answer  equally  well  for  a  superior  planet,  provided 
we  regard  m  as  standing  for  the  mean  daily  motion  of  the  earth, 
M  for  that  of  the  planet,  p  for  the  sidereal  period  of  the  earth, 
and  P  for  that  of  the  planet.  For  the  earth  holds  towards  a 
superior  planet  the  place  of  an  inferior  planet,  and  a  synodic 
revolution  of  the  earth,  to  an  observer  on  the  planet,  will  ob- 
viously be  a  synodic  revolution  of  the  planet  to  an  observer  on 
the  earth. 

349.  Lengths    of    Synodic     Revolution*    off    PlaiieU. 
Equa.  (30)  shows  that  the  length  of  a  mean  synodic  revolution 
depends  altogether  upon  the  amount  of  the  difference  of  the 
mean  daily  motions  of  the  earth  and  planet,  and  is  the  greater 
the  less  is  this  difference. 

It  follows,  therefore,  that  the  synodic  revolution  is  the  longest 
for  the  planets  nearest  the  earth. 

It  appears  by  equa.  (81)  that  the  length  of  a  synodic  revo- 
lution is,  for  an  inferior  planet,  greater  than  the  sidereal  period 
of  the  planet,  and,  for  a  superior  planet,  greater  than  the  sidereal 
period  of  the  earth.  The  actual  lengths  of  the  synodic  revolu- 
tions of  the  different  planets  are  given  in  Table  V. 

The  mean  synodic  revolution  of  a  planet  being  known,  and 
also  the  time  of  one  conjunction  of  opposition,  we  may  easily 
ascertain  its  mean  elongation  at  any  given  time,  and  thus  approx- 
imately the  time  of  its  rising,  setting,  and  meridian  passage. 

350.  Planet§  as  Evening  or  Morning  Stars.     A  planet 
will  rise  and  set  at  the  same  hours  at  the  end  of  a  synodic  revo- 
lution; and  will  be  an  evening  star,  that  is,  above  the  horizon 
at  sunset,  during  half  of  a  synodic  revolution,  and  a  morning 
star,  that  is,  above  the  horizon  at  sunrise,  during  an  equal  inter- 
val of  time.     The  inferior  planets  will  be  evening  stars  from 
superior  to  inferior  conjunction ;  and  the  superior  planets  from 
opposition  to  conjunction. 

Mercury  is  an  evening  star  for  a  period  of  2  months  ;  Venus 
during  an  interval  of  9?  months ;  Mars  for  1  year  and  1  month  ; 
Jupiter  for  6£  months ;  Saturn  and  Uranus  each  a  few  days  more 
than  6  months. 


STATIONS  AND  RETROGRADATIONS  OF  THE  PLANETS. 

351.  The  apparent  motions  of  the  planets  in  the  heavens,  as 
•has  already  been  stated  (11),  are  not,  like  those  of  the  sun  and 
moon,  continually  from  west  to  east,  or  direct,  but  are  sometimes 
also  from  east  to  west,  or  retrograde.  The  retrograde  motion 
takes  place  over  arcs  of  but  a  small  number  of  degrees ;  and  in 
changing  the  direction  of  their  motions,  the  planets  are  for  several 
days  stationary  in  the  heavens.  These  phenomena  are  called  the 
Stations  and  Retrogradations  of  the  planets.  We  now  propose  to 


212 


THE  PLANETS. 


inquire  theoretically  into  the  particulars  of  the  motions  in  ques- 
tion, and  to  show  how  the  phenomena  just  mentioned  result 
from  the  motions  of  the  planets  in  connection  with  the  motion  of 
the  earth. 

352.  Case  of  an  Inferior  Planet.  Let  CAC'B  (Fig.  85) 
represent  the  orbit  of  an  inferior  planet,  and  PKT  the  orbit  of 
the  earth ;  both  considered  as  circles,  and  as  situated  in  the  same 
plane.  If  the  earth  were  continually  stationary  in  some  point  P 
of  its  orbit,  it  is  plain  that  while  the  planet  was  moving  from  B 
the  position  of  greatest  western  elongation  to  A  the  position  of 
greatest  eastern  elongation,  it  would  advance  in  the  heavens  from 
b  to  a ;  that,  while  it  was  moving  from  A  to  B,  that  is,  from 
greatest  eastern  to  greatest  western  elongation,  it  would  retro- 
grade in  the  heavens  from  a  to  b  ;  and  that,  in  passing  the  points 
A  and  B,  as  it  would  be  moving  directly  towards  or  from  the 
earth,  it  would  for  a  time  appear  stationary  in  the  heavens,  in 
the  positions  a  and  b. 

But  the  earth  is  in.  fact  in  motion,  and  the  actual  apparent 
motion  of  the  planet  is  in  consequence  materially  different  from 
this.  Let  A,  A'  (Fig.  86)  be  the  positions  of  the  planet  and 
earth,  at  the  time  of  the  greatest  eastern  elongation,  C',  P  their 
positions  at  inferior  conjunction,  and  B,  B'  their  positions  at  the 


FIG.  86. 


greatest  western  elongation.  At  the  time  of  the  greatest  eastern 
elongation,  while  the  planet  describes  a  certain  distance  AD  on 
the  line  of  the  centres  of  the  earth  and  planet,  the  earth  moves  for- 


STATIONS  AND  RETROGRADATIONS  OF  THE   PLANETS.     21 3 

ward  in  its  orbit  a  certain  distance  A'D';  so  that,  instead  of 
appearing  stationary  at  a  in  the  interval,  the  planet  will  advance 
in  the  heavens  from  a  to  d.  From  the  same  cause  it  will  have  a 
direct  motion  about  the  time  of  the  greatest  western  elongation. 
As  it  advances  from  A  towards  C',  the  direct  motion  will  con- 
tinue ;  but,  as  the  daily  arc  described  by  the  planet  will  make  a 
less  and  less  angle  with  the  daily  arc  described  by  the  earth,  the 
rate  of  motion  will  continually  decrease,  and  finally,  when  the 
planet  has  come  into  a  position  with  respect  to  the  earth,  such 
that  the  lines  of  direction  of  the  planet,  mp.  m'p ',  at  the  begin- 
ning and  end  of  the  day  are  parallel,  it  will  be  stationary  in  the 
heavens.  As  the  daily  arc  of  the  planet  is  greater  than  that  of 
the  earth,  and  becomes  parallel  to  it  in  inferior  conjunction,  the 
planet  will  be  in  the  position  in  question  before  it  comes  into 
inferior  conjunction. 

Subsequent  to  this,  the  inclination  of  the  daily  arcs  still  dimin- 
ishing, the  lines  of  direction  of  the  planet  at  the  beginning  and 
end  of  the  day  will  diverge,  and  therefore  the  motion  will  be  re- 
trograde. After  inferior  conjunction,  the  inclination  of  the  arcs 
will,  at  corresponding  positions  of  the  earth  and  planet,  obviously 
be  the  same  as  before.  It  follows,  therefore,  that  the  planet  will 
be  at  its  western  station  when  it  is  at*  the  same  angular  distance 
from  the  sun  as  at  its  eastern  station ;  that  its  motion  will  be 
retrograde  until  it  has  passed  inferior  conjunction  and  arrived  at 
its  western  station  ;  and  that  after  this  it  will  be  direct,  q  and  n 
represent  the  positions  of  the  planet  and  the  earth  at  the  time  of 
the  western  station ;  C'q  =  C'p,  and  Pn=Pm. 

The  diminution  of  the  elongation  of  the  planet  at  its  two  sta- 
tions is  not  the  only  effect  of  the  earth's  motion  in  the  case  under 
consideration  ;  it  also  accelerates  the  direct,  and  retards  the  retro- 
grade motion  of  the  planet,  and  gives  to  the  planet  along  with 
the  sun  an  apparent  motion  of  revolution  around  the  earth. 

353.  Case  of  a  Superior  Planet.  Suppose  AC'B  (Fig. 
86)  to  be  the  orbit  of  the  earth,  and  A'PB'  that  of  the  planet. 
Since  the  earth  is  an  inferior  planet  to  an  observer  stationed  upon 
a  superior  planet,  it  appears  by  the  foregoing  article  that  it  will, 
to  an  observer  so  situated,  have  a  retrograde  motion  while  it  is 
passing  over  a  certain  arc  pC'q  in  the  inferior  part  of  its  orbit, 
and  a  direct  motion  during  the  remainder  of  the  synodic  revolu- 
tion. Now,  it  is  plain  that  the  direction  of  the  planet's  motion, 
as  seen  from  the  earth,  will  always  be  the  same  as  the  direction 
of  the. earth's  motion  as  seen  from  the  planet.  When  the  earth 
is  at  C',  the  middle  of  the  arc  pG'q,  the  planet  is  in  opposition. 
It  follows,  therefore,  that  a  superior  planet  has  a  retrograde  mo- 
tion during  a  small  portion  of  its  synodic  revolution,  about  the 
time  of  opposition.  (See  Table  Y.) 


214: 


THE   PLANETS. 


o 


PHASES  OF  THE  INFERIOR  PLANETS. 

354.  To  the  naked  sight  the  disc  of  the  planet  Venus  appears 
circular,  like  that  of  each  of  the  other  planets,  but  the  telescope 
shows  this  to  be  an  optical  illusion.     When  Yenus  is  repeatedly 
observed  with  a  telescope,  it  is  seen  to  present  in  its  various  posi- 
tions with  respect  to  the  sun  the  same  variety  of  phases  as  the 
moon  ;  being  a  full  circle  at  superior  conjunction,  a  half  circle  at 
the  greatest  eastern  and  western  elongations,  and  a  crescent,  with 
the  horns  turned  from  the  sun,  before  and  after  inferior  con- 
junction. 

Mercury  exhibits  precisely  similar  phases,  but  being  smaller,  at 
a  greater  distance  from  the  earth,  and  much  nearer  the  sun,  its 
phases  are  not  so  easily  observed  as  those  of  Yenus. 

355.  Explanation.    The  phases  of  Yenus  are  easily  account- 
ed for,  by  supposing  it  to  be  an  opake  spherical  body,  and  to 
shine  by  reflecting  the  sun's  light,  and  by  taking  into  considera- 
tion  its   motion  with   respect   to 
the  sun   and  earth.     The   hemi- 
sphere  turned    towards  the   sun 
is   illuminated  and   the   other  is 
in  the   dark,    and  as  the  planet 
revolves  around  the  sun,  various 
portions  of  the  enlightened  half 
are  turned  towards  the  earth  ;  in 
superior  conjunction,  the  whole  of 
it ;  at  the  greatest  elongations,  one 
half;  and  near  inferior  conjunc- 
tion, but  a  small  part.     This  will 
be  abundantly  evident  on  inspect- 
ing Fig.  87.     The  phases  corres- 
ponding to  the  positions  represent- 
ed are  delineated  in  the  figure. 

The  phases  of  Mercury  are  ob- 
viously suceptible  of  a  similar  ex- 
planation. 

356.  Changes  of  Form  of  the  Di§c  of  Mars.      The    disc 
of  the  planet  Mars  also  undergoes  changes  of  form,  but  they  are 
of  comparatively  moderate  extent.     It  is  sometimes  gibbous,  but 
never  has  the  form  of  a  crescent.     Indeed,  on  the  supposition 
that  Mars  is  an  opake  body  illuminated  by  the  sun,  we  would 
not  see  the  whole  of  the  enlightened  hemisphere,  except  in  con- 
junction and  opposition,  but  there  would  always  be  more  than 
half  of  it  turned  towards  the  earth,  and  therefore  the  disc  should 
always  be  larger  than  a  half  circle. 

The  discs  of  the  other  superior  planets  do  not  experience  any 
perceptible  variation  of  form,  for  the  reason,  doubtless,  that  their 


FIG.  87. 


TRANSITS   OF  THE   INFERIOR  PLANETS.  215 

orbits  are  so  large  with  respect  to  the  orbit  of  the  earth,  that  all, 
or  very  nearly  all  of  their  illuminated  hemispheres,  is  constantly 
visible  from  the  earth.  Jupiter  offers  the  only  exception  to  this 
general  truth ;  it  is  slightly  gibbous  in  quadratures. 


TRANSITS  OF  THE  INFERIOR  PLANETS. 

.  The  two  inferior  planets  Venus  and  Mercury,  at  inferior 
conjunction,  sometimes,  though  rarely,  pass  between  the  sun  and 
earth,  and  are  seen  as  a  dark  spot  crossing  the  sun's  disc.  This 
phenomenon  is  called  a  Transit.  It  will  take  place,  in  the  case 
of  either  planet,  whenever,  at  the  time  of  inferior  conjunction,  it 
is  so  near  either  node  that  its  geocentric  latitude  is  less  than  the 
apparent  semi-diameter  of  the  sun. 

'       35§.    Epochs    of    Transits :— Periods    of    Recurrence. 

£  The  transits  of  Venus  take  place  alternately  at  intervals  of  8  and 
105J  or  121-$-  years.  The  last  were  in  the  years  1761  and  1769. 
The  next  will  be  in  1874  and  1882  ;  of  which  the  latter  will  be 
visible  in  this  country. 

In  consequence  of  the  greater  distance  of  Mercury  from  the 
earth,  a  greater  portion  of  its  orbit  is  directly  interposed  between 
the  sun  and  earth  than  of  the  orbit  of  Venus ;  moreover,  the 
synodic  revolution  of  Mercury  is  shorter  than  that  of  Venus. 
On  these  accounts  it  happens  that  the  tram  sits  of  Mercury  are 
much  more  frequent  than  those  of  Venus.  The  last  transit  of 
Mercury  was  on  November  11,  1861.  The  next  two  will  take 
place  in  1868  and  1878,  in  November  and  May.  The  first, 
which  will  occur  on  the  4th,  will  be  visible  in  this  country. 

359.  A  Transit  i§  Calculated  in  a  precisely  similar  man- 
ner with   a  solar   eclipse ;    the  planet  in    the  one  calculation 
answering  to  the  moon  in  the  other. 

360.  A   Transit   is    an   Important   Phenomenon  in    a 
practical  point  of  view,  as  it  furnishes  an  indirect  but  accurate 
method  of  ascertaining  the  sun's  parallax.     In  order  to  under- 
stand how  this  phenomenon   can  be  used  for  this  purpose,  we 
have  only  to  consider  that,  in  consequence  of  the  difference  of 


FIG.  88. 


the  distances  or  parallaxes  of  the  sun  and  Venus,  observers  at 
different  stations  upon  the  earth  will  refer  the  planet  to  different 
points  upon  the  sun's  disc,  and  that  therefore,  to  such  observers, 


216  THE   PLANETS. 

the  transit  will  take  place  along  different  chords,  and  be  accom- 
plished in  unequal  portions  of  time.  This  fact  is  represented  to 
the  eye  in  Fig.  88.  It  is  then  to  be  expected,  that,  if  the  dura- 
tions of  the  transit  at  two  different  places  should  be  noted,  the 
difference  of  the  parallaxes  of  the  sun  and  Venus,  upon  which 
alone  the  difference  of  the  durations  depends,  could  be  computed. 
This  computation  is  in  fact  possible.  Also,  the  ratio  of  the  par- 
allaxes being  inversely  as  that  of  the  distances,  could  be  found 
by  the  elliptic  theory  of  the  planetary  motions,  and  thus  the 
parallax  both  of  the  sun  and  Venus  would  become  known. 

361.  The  Parallax  of  the  Sim  was  quite  accurately  de- 
duced from  observations  upon  the  transits  of  Venus  in  1769  and 
1761.     Expeditions  were  fitted  out  on  the  most  efficient  scale,  by 
the  British,  French,  Russian,  and  other  governments,  and  sent  to 
various  parts  of  the  earth,  remote  from  each  other,  to  observe 
the  transit  of  1769,  that  the  parallax  of  the  sun  might  be  com- 
puted from  the  results  of  the  observations.     The  sun's  parallax, 
as  determined  by  Professor  Encke  from  the  observations 'made 
upon  the  transit  in  question,  and  that  of  1761,  is  S".5776.     We 
have  already  seen  that  the  sun's  parallax  has  recently  been  more 
accurately  determined  (150). 

APPEARANCE,  DIMENSIONS,  ROTATION,  AND  PHYSICAL 
CONSTITUTION  OF  THE  PLANETS. 

362.  Variation*  of  Apparent  Diameter.  It  appears  from 
admeasurement  with  the  telescope  and  micrometer,  that  the  ap- 
parent diameter  of  a  planet  is  subject  to  sensible  variations.     The 
apparent  diameter  of  Venus,  as  well  as  of  Mercury,  is  greatest 
in  inferior  conjunction,  and  least  in  superior  conjunction  ;  while 
the  apparent  diameter  of  each  of  the  other  planets  is  greatest  in 
opposition  and  least  in  conjunction.     These  variations  of  the  ap- 
parent diameters  of  the  planets  are  necessary  consequences  of  the 
changes  that  take  place  in  the  distances  of  the  planets  from  the 
earth.     (See  Fig.  85.) 

363.  Afogolnte  and  Relative  Magnitudes     The  real  dia- 
meter of  a  planet  is  deduced  from  its  apparent  diameter  and 
horizontal  parallax.     (See  Art.  310.)     When  the  diameters  of 
the  planets  have  been  found,  their  relative  surfaces  and  volumes 
are  easily  obtained ;  for  the  surfaces  are  as  the  squares  of  the 
diameters,  and  the  volumes  as  the  cubes* 

The  order  of  magnitude  of  the  planets  is  as  follows  :  1  Jupiter, 
2  Saturn,  3  Neptune,  4  Uranus,  5  the  Earth,  6  Venus,  7  Mars, 
8  Mercury,  9  Pallas,  10  Vesta,  11  Ceres,  12  Juno,  13  the  other 
planetoids.  The  range  of  magnitude,  for  the  principal  planets, 
is  from  1  to  about  25,000.  The  relative  magnitudes  of  the  princi- 
pal planets  are  given  in  Table  IV. 

364.  Rotation  of  Planets.     Spots  more  or  less  dark  have 


MERCURY.  217 

been  seen  upon  the  discs  of  most  of  the  principal  planets ;  and 
by  passing  across  them  from  east  to  west  and  reappearing  at  the 
eastern  limbs,  have  established  that  the  planets  upon  which  they 
a/e  observed  rotate  upon  axes  from  west  to  east.  From  repeated 
careful  observations  upon  the  situations  of  these  spots,  the  periods 
of  rotation,  and  the  positions  of  the  axes,  have  been  determined. 

The  periods  of  rotation  of  Mercury,  Venus,  the  Earth,  and 
Mars,  are  all  about  24  hours,  and  of  Jupiter  and  Saturn  about 
10  hours.  Those  of  the  other  planets  are  not  known.  The 
axes  of  rotation  remain  continually  parallel  to  themselves,  as  the 
planets  revolve  in  their  orbits. 

365.  The  Amount  of  Light  and  Heat,  which  the  sun 
bestows  upon  the  planets  decreases  in  the  same  ratio  that  the 
square  of  the  distance  increases.  (See  Table  IV.) 

It  will  be  seen  in  the  sequel  that  the  planets  are  all  opake 
bodies,  like  the  earth,  and  shine  wholly  by  the  reflected  light  of 
the  sun ;  and  that  most,  if  not  all  of  them,  are  surrounded  with 
atmospheres. 


MERCURY. 

306.  In  consequence  of  its  proximity  to  the  sun,  Mercury  is 
rarely  visible  to  the  naked  eye.  When  seen  under  the  most 
favorable  circumstances,  about  the  time  of  greatest  elongation, 
and  at  periods  of  the  year  when  twilight  is  of  short  duration,  it 
presents  the  appearance  of  a  star  of  the  third  or  fourth  magni- 
tude. Its  phases  indicate  that  it  is  opake  and  illuminated  by  the 
sun.  Its  apparent  diameter  varies  with  its  distance  from  the 
earth,  from  5"  to  13".  Its.  real  diameter  is  a  little  less  than  3,000 
miles,  or  -£•  of  tha,t  of  the  earth ;  and  its  volume  is  about  ^  of 
the  earth's  volume.* 

Mercury  performs  a  complete  rotation  on  its  axis  in  24h.  5£m., 
and  according  to  Schroter,  its  axis  is  inclined  to  the  plane  of  the 
ecliptic  under  as  small  an  angle  as  20°. 

367.  Telescopic  Appearances.  Owing  to  the  dazzling  splendor  of  its 
light,  and  the  tremulous  motion  induced  by  the  ever-varying  density  of  the  air 
and  vapors  near  the  earth's  surface,  through  which  it  is  seen,  the  telescope  does 
not  present  a  well-defined  image  of  the  disc  of  this  planet.  Scbroter  is  the  only 
observer  who  has  supposed  that  he  discerned  distinct  spots  upon  it  Later  observ- 
ers have  only  noticed  on  rare  occasions,  slight  inequalities  of  brightness  on  its 
disc. 

From  the  fact  that  such  appearances  are  only  occasionally  seen,  it  has  been  in- 
ferred that  the  planet  is  surrounded  with  a  dense  atmosphere  loaded  with  clouds, 
that  reflect  a  strong  light,  and,  except  when  the  atmosphere  clears  up  in  an  un- 
usual degree,  prevent  the  darker  body  of  the  planet  from  being  seen.  But  the 
evidence  in  support  of  this  conclusion  needs  confirmation. 

Schroter,  in  making  observations  upon  Mercury  at  the  time  the  disc  had  the 
form  of  a  crescent,  discovered  that  one  of  the  horns  of  the  crescent  became  blunt  at 

*  The  exact  diameters,  volumes,  times  of  rotation,  &c.,  of  the  different  planets,  as 
*ar  as  known,  may  be  found  in  Table  IY. 


218  THE   PLANETS. 

the  end  of  every  24  hours ;  from  which  he  inferred  that  the  planet  turned  upon  an 
axis,  and  had  mountains  upon  its  surface,  which  were  brought  at  the  end  of  every 
rotation  into  the  same  position  with  respect  to  his  eye  and  the  sun. 

YENUS. 

36§.  Venus  is  the  brightest  of  all  the  planets,  and  generally 
appears  larger  and  brighter  than  any  of  the  fixed  stars.  But  it 
is  much  more  conspicuous  at  some  times  than  at  others,  during  a 
synodic  revolution.  It  is  found  by  calculation,  that  the  epochs 
in  the  course  of  a  synodic  revolution  at  which  Yenus  gives  most 
light  to  the  earth,  are  those  at  which,  being  in  the  inferior  part 
of  its  orbit,  it  has  an  elongation  from  the  sun  of  a. little  less  than 
40°.  The  disc  is  then  less  than  one-quarter  of  a  circle,  but  the 
increased  proximity  to  the  earth  more  than  compensates  for  the 
diminished  size  of  the  disc.  Yenus  attains  to  greater  splendor  in 
some  revolutions  than  in  others,  in  consequence  of  being  nearer 
the  earth  when  in  the  favorable  position  just  noticed.  A  com- 
bination of  the  most  favorable  circumstances  recurs  every  eight 
years,  when  Yenus  becomes  visible  in  full  daylight,  and  casts  a 
sensible  shadow  at  night.  This  last  happened  in  February,  1862. 

As  seen  through  a  telescope,  Yenus  presents  a  disc  of  nearly 
uniform  brightness,  and  spots  have  very  rarely  been  seen  upon 
it.  From  the  regular  succession  of  phases  through  which  the 
disc  passes,  as  the  planet  changes  its  position  with  respect  to  the 
earth  and  sun,  we  infer  that  it  is  an  opake  spherical  body,  shin- 
ing by  the  reflected  light  of  the  sun.  Its  apparent  diameter 
varies  with  its  distance  from  the  earth,  from  10"  to  66".  Its  real 
diameter  is  7,600  miles ;  and  its  volume  %  less  than  that  of  the 
earth.  The  period  of  its  rotation  is  23h.  21m.  The  inclination 
of  its  axis  to  the  plane  of  its  orbit  is  not  exactly  known,  but 
is  supposed  to  be  not  far  from  18°. 

369.  Evidences  of  an  Atmosphere  surrounding  Venus.  From 

the  remarkable  vivacity  of  the  light  of  this 
planet,  which  far  exceeds  that  of  the  light 
reflected  from  the  moon's  surface,  as  well  as 
the  transitory  nature  of  the  few  darkish  spots 
that  have  been  seen  upon  its  disc,  it  is  inferred 
that  it  is  surrounded  by  a  dense  and  highly 
reflective  atmosphere,  which  in  general  screens 
the  whole  of  the  darker  body  of  the  planet 
from  view.  The  truth  of  this  inference  is 
confirmed  by  certain  delicate  observations 
made  by  Schroter.  This  Astronomer  distinct- 
ly discerned,  when  the  disc  was  seen  as  a  nar- 
row crescent,  a  faint  light  stretching  beyond 
the  proper  termination  of  one  of  the  horns  of 
the  crescent  into  the  dark  part  of  the  face  of 
the  planet,  as  is  represented  in  Fig.  89,  where 
the  left  extremity  of  the  dotted  line  represents 
FIG.  89.  the  natural  terminating  point  of  one  of  the 

horns. 

The  same  appearance  has  since  been  repeatedly  noticed  by  other  observers     It 


FlGK  61. 


65. 


TENUS.  219 

was  distinctly  perceptible  before  and  after  the  last  inferior  conjunction  of  the  planet, 
December  llth,  1866.  The  planet  was  watched  from  day  to  day  by  Professor  Ly- 
man,-  with  the  nine-inch  equatorial  of  the  Sheffield  Observatory,  Yale  College,  until, 
on  the  day  before  conjunction,  its  distance  from  the  nearest  limb  of  the  sun  was  only 
1°  8'.  The  very  slender  crescent  which  it  exhibited,  was  each  day  seen  more  and 
more  extended  beyond  a  semicircle ;  until  at  favorable  moments,  when  the  sun, 
but  not  the  planet,  was  covered  by  a  passing  cloud,  it  was  distinctly  observed  as 
an  entire  ring  of  light,  thinnest  on  the  side  furthest  from  the  sun.  The  entire 
ring  was  seen  also,  by  the  same  observer,  with  a  five-foot  telescope,  so  placed  as 
to  have  the  sun  covered  by  a  distant  chimney.  The  maximum  extent  of  the  cres- 
cent observed  at  Dorpat,  at  the  conjunction  in  1849,  was  240 3;  the  planet  being 
3'  26'  from  the  sun's  centre. 

This  remarkable  prolongation  of  the  cusps  must  be  attributed  mainly  to  the  re- 
fraction of  the  sun's  rays  by  the  atmosphere  of  the  planet.  On  this  assumption, 
Miidler,  from  the  Dorpat  observations  of  the  extent  of  the  cusps,  made  the  hori- 
zontal atmospheric  refraction  of  Venus  43'.7.  The  observations  of  Professor  Lyman, 
at  the  late  conjunction,  give  45'.3.  This  is  about  ^  greater  than  the  horizontal 
refraction  produced  by  the  earth's  atmosphere ;  and  indicates  that  the  density  of 
the  atmosphere  of  Venus  is  decidedly  greater  than  that  of  the  earth's  atmosphere. 

370.  C-ouds   in   the  Atmosphere.     Since  the  transparency  of  Venus's  atmos- 
phere   is    variable,  becoming  occasionally  such    as   to    admit   of   the  body  of 
the  planet's   being  seen  through  it,   we  must  suppose  that    it  contains  aque- 
ous vapor  and   clouds,   and  therefore   that  there  are  bodies  of  water  upon  the 
surface  of  the  planet.     It  is  in  fact  supposed  that  isolated  clouds  have  actually 
been  seen.     The  most  natural  explanation  of  the  bright  spots  which  have  some- 
times been  noticed  on  the  disc,  is,  that  they  are  clouds,  more  highly  reflective  than 
the  atmosphere,  or  than  the  clouds  in  general. 

371.  Inequalities  on  the   Surface.    There   are  great  inequalities  on  the  sur- 
face of  Venus,  and,  it  would  seem,  mountains,    much  higher    than  any  upon 
our  globe.     Schroter  detected  these  masses  by  several  infallible  marks.    In  the 
first  place,  the  edge  of  the  enlightened  part  of  the  planet  is  shaded,  as  seen  in 
Figs.  89,  90,  91,  and  as  the  moon  appears  when  in  crescent  even  to  the  naked  eye. 


FIG.  90.  JIG.  91. 


This  appearance  is  doubtless  caused  by  shadows  cast  by  mountains ;  which  are 
naturally  best  seen  on  that  part  of  the  planet  to  which  the  sun  is  rising  or  setting, 
where  they  are  longest.  In  the  next  place,  the  edge  of  the  disc  shows  marked 
irregularities.  Thus  it  sometimes  appears  rounded  at  the  corners,  as  in  Fig.  90, 
owing  undoubtedly  to  part  of  the  disc  being  rendered  invisible  there  by  the  shadow 
or  interposition  of  some  line  of  eminences ,  and  at  other  times,  as  in  Fig.  91,  a 
single  bright  point  appears  detached  from  the  disc— the  top  of  a  high  mountain, 
illuminated  across  a  dark  valley. 

Schroter  found  that  these  appearances  recurred  regularly,  at  equal  intervals  of 
about  23£  hours ;  the  same  period  as  that  which  Cassini  had  previously  found  for 
the  completion  of  a  rotation,  by  observations  upon  the  spots. 


220  THE  PLANETS.  . 

MARS. 

372.  Mars  is  of  the  apparent  size  of  a  star  of  the  first  or 
'second  magnitude,  and  is  distinguished  from  the  other  planets 
"by  the  ruddy  color  of  its  light.  The  observed  variation  in  the 
form  of  its  disc  (356)  shows  that  it  derives  its  light  from  the  sun. 
Its  greatest  and  least  apparent  diameters  are  respectively  4"  and 
80/y.  Its  real  diameter  is  somewhat  less  than  5,000  miles,  and 
its  bulk  is  about  i  of  that  of  the  earth. 

Mars  revolves  on  its  axis  in  24h.  37m. ;  and  its  axis  is  in- 
clined to  the  plane  of  the  ecliptic  in  an  angle  of  about  60°.  It 
appears,  from  measurements  made  with  the  micrometer,  that  its 
polar  diameter  is  less  than  the  equatorial,  and  thus,  that  like  the 
earth,  it  is  flattened  at  its  poles.  According  to  •  the  latest  deter- 
minations its  oblateness  (105)  is  -fa. 

3t3.  Tele§copic  Appearaiice§  : — Inferences^  When  the 
disc  of  Mars  is  examined  with  telescopes  of  great  power,  it  is 
generally  seen  to  be  diversified  with  large  spots  of  different 
shades,  which,  with  occasional  variations,  retain  constantly  the 
same  size  and  form. 

These  are  conjectured  to  be  continents  and  seas.    In  fact,  Sir  J.  F.  W.  Herschel 

has  on  several  occasions,  in  exam- 
ining this  planet  with  a  good  tele- 
scope, noticed  that  some  of  its 
spots  are  of  a  dull  red  color,  while 
others  have  a  greenish  hue.  The 
former  he  supposes  to  be  land, 
and  the  latter  water.  Fig.  92 
represents  Mars  in  its  gibbous 
state,  as  seen  by  Herschel  in  his 
twenty-feet  reflector,  on  the  16th 
of  August,  1830.  The  darker 
parts  are  the  supposed  seas.  The 
bright  spot  at  the  top  is  at  one 
of  the  poles  of  Mars.  At  other 
times  a  similar  bright  spot  is 
seen  at  the  other  pole.  These 
brilliant  white  spots  have  been 
conjectured,  with  a  great  deal  of 
probability,  to  be  snow;  as  they 
FIG.  92  are  reduced  in  size,  and  some- 

times disappear  when  they  have 

been  long  exposed  to  the  sun,  and  are  greatest  when  just  emerging  from  the  long 
night  of  their  polar  winter. 

The  great  divisions  of  the  surface  of  Mars  are  seen  with  different  degrees  of 
distinctness  at  different  times,  and  sometimes  disappear,  either  partially  or  entire- 
ly; parts  of  the  disc  also  appear  at  times  particularly  dark  or  bright.  From  these 
facts  it  is  to  be  inferred  that  this  planet  is  environed  with  an  atmosphere,  and 
that  this  contains  aqueous  vapor,  which,  by  varying  in  quantity  and  density, 
renders  its  transparency  variable. 

37-4.  No  nwuntains  have  been  detected  upon  Mars.  But  this  is  no  good  reason 
for  supposing  that  they  are  really  wanting  there ;  for,  if  the  surface  of  Mars  be 
actually  diversified  with  mountains  and  valleys,  since  its  disc  never  differs  much 
from  a  full  circle,  we  have  no  reason  to  expect  that  its  edge  would  present  that 
Bhaded  appearance  and  those  irregularities  which  have  been  noticed  on  Venus 


JUPITER   AND   ITS   SATELLITES. 


221 


and  Mercury,  when  of  the  form  of  a  crescent.     The  same  remarks  will  apply,  with 
still  greater  force,  to  the  other  superior  planets. 

375.  The  ruddy  color  of  the  light  of  Mars  has  generally  been  attributed  to  its 
atmosphere,  but  Sir  John  Herschel  finds  a  sufficient  cause  for  this  phenomenon 
in  the  ochrey  tinge  of  the  general  soil  of  the  planet. 


JUPITER  AND  ITS  SATELLITES. 

376.  Jupiter  is  the  most  brilliant  of  the  planets,  except  Yenus, 
and  sometimes  even  surpasses  Yenus  in  brightness.     The  general 
fact  and  special  circumstances  of  the  eclipses  of  its  satellites,  and 
of  the  transits  of  the  shadows  across  the  disc  of  their  primary 
(243),  indicate  that  Jupiter,  as  well  as  its  satellites,  are  opake 
bodies,  and  shine  by  the  reflected  light  of  the  sun.     Its  apparent 
diameter,  when  greatest,  is  51",  and  when  least,  3 1". 

Jupiter  is  the  largest  of -all  the  planets ;  its'equatorial  diameter 
is  11  times  that  of  the  earth,  or  88,000  miles,  and  its  bulk  is  very 
nearly  1,300  times  that  of  the  earth.  It  turns  on  an  axis  nearly 
perpendicular  to  the  ecliptic,  and  completes  a  rotation  in  9h. 
o.Hrn-  The  polar  diameter  is  -^  less  than  the  equatorial. 

377.  Belt*  of  Jupiter.     When  Jupiter  is  examined  with  a 
good  telescope,  its  disc  is  always  observed  to  be  crossed  by  several 
obscure  spaces,  which  are  .nearly  parallel  to  each  other  and  to  the 
equator.     These  are  called  the  Belts  of  Jupiter  (see  Fig.  93; 
which  represents  the  appear- 
ance of  Jupiter  as  seen  by 

Sir  John  Herschel,  in  his 
twenty-feet  reflector,  on  the 
23d  of  September,  1832.) 
They  vary  somewhat  in 
number,  breadth,  and  situa- 
tion on  the  disc,  but  never 
in  direction.  Sometimes 
only  one  or  two  are  visible ; 
on  other  occasions  as  many 
as  eight  have  been  seen  at 
the  same  time.  Sir  William 
Herschel  eve'n  saw  them,  on 
one  or  two  occasions;  broken 
up  and  distributed  over  the  PIG.  93. 

whole  face  of  the   planet; 

but  this  phenomenon  is  extremely  rare.  Branches  running  out 
from  the  belts,  and  subdivisions,  as  represented  in  the  figure,  are 
by  no  means  uncommon.  Dark  Spots  of  invariable  form  and 
size  have  also  been  occasionally  seen  upon  them.  These  have 
been  observed  to  have  a  rapid  motion  across  the  disc,  and  to 
return  at  equal  intervals  to  the  same  position  on  the  disc,  after 
the  same  manner  as  the  sun's  spots ;  which  leaves  no  room  to 


222  THE   PLANETS. 

doubt  that  they  are  on  the  body  of  the  planet,  and  that  this  turns 
upon  an  axis.  Bright  Spots  have  also  recently  been  detected 
upon  the  belts  by  two  observers ;  Dawes  and  Lassell.  The  belts 
generally  retain  pretty  nearly  the  same  appearance  for  several 
months  together,  but  occasionally  marked  changes  of  form  and 
size  take  place  in  the  course  of  aa  hour  or  two.  They  are  even 
said  to  change  sometimes  very  sensibly  in  the  course  of  a  few 
minutes. 

Explanation  of  the  Belts.  The  occasional  variations  of  Jupiter's  belts,  and  the 
occurrence  of  spots  upon  them,  which  are  undoubtedly  permanent  portions  of  the 
mass  of  the  planet,  render  it  extremely  probable  that  they  are  the  body  of  the 
planet  seen  through  an  atmosphere  of  variable  transparency,  but,  in  general, 
having  extensive  tracts  of  comparatively  clear  sky  in  a  direction  parallel  to  the 
equator.  These  are  supposed  to  be  determined  by  currents  analogous  to  our  trade- 
winds,  but  of  a  much  more  steady  and  decided  character,  as  would  be  the  neces- 
sary consequence  of  the. superior  velocity  of  rotation  of  this  planet.  As  remarked 
by  Herschel,  that  it  is  the  comparatively  dark  body  of  the  planet  which  appears 
in  the  belts,  is  evident  from  this,  that  they  do  not  come  up  in  all  their  strength  to 
the  edge  of  the  disc,  but  fade  away  gradually  before  they  reach  it.  The  bright 
belts,  intermediate  between  the  dark  ones,  are  believed  to  be  bands  of  clouds,  or 
tracts  of  less  pure  air. 

It  is  possible  that  these  bright  belts  may  be  of  the  nature  of  auroral  rather  than 
aqueous  clouds,  and  that  the  dark  belts  may  result  from  their  dispersion  along 
certain  tracts,  the  process  being  controlled  by  the  varying  operation  of  the  sun 
and  planets :  after  the  manner  that  the  planets  operate  upon  the  photosphere  of 
the  sun,  to  develop  spots  upon  the  sun's  disc.  Such  clouds  may  have  a  certain 
degree  of  luminosity,  and  yet  at  the  distance  of  the  earth  may  shine  by  the  reflect- 
ed light  of  the  sun.  The  general  prevalence  of  dark  belts  on  either  side  of  the 
equator,  separated  by  a  bright  band  at  the  equator,  is  analogous  to  the  two  spot- 
belts  of  the  sun,  with  an  intervening  region  from  which  the  spots  are  absent. 

If,  as  has  boeii  maintained  by  the  author  in  other  publications,  the  collision  of 
the  particles  of  the  earth  and  planets  with  the  ether  of  space  develops  heat,  not 
only  directly,  but  by  the  origination  of  electric  currents  which  subsequently  pass 
off  in  the  form  of  heat,  then  since  a  point  on  the  equator  of  Jupiter  has  a  rotatory 
velocity  28  times  greater  than  that  of  a  point  on  the  equator  of  the  earth,  the  tem- 
perature at  the  surface  of  Jupiter  may  be  much  greater  than  that  of  the  earth, 
notwithstanding  its  greater  distance  from  the  sun.  Upon  this  idea,  it  is  natural 
to  expect  a  certain  degree  of  similarity  in  the  photospheric  condition  of  this  planet 
and  the  sun. 

37§.  The  Satellites  of  Jupiter,  as  it  has  been  already  re- 
marked, are  visible  with  telescopes  of  very  moderate  power. 
With  the  exception  of  the  second,  which  is  a  little  smaller,  they 
are  a  little  larger  than  the  moon.  The  orbits  of  the  satellites  lie 
very  nearly  in  the  plane  of  Jupiter's  equator.  They  are,  there- 
fore, viewed  nearly  edgewise  from  the  earth,  and  in  consequence 
the  satellites  always  appear  nearly  in  a  line  with  each  other. 

Sir  W.  Herschel,  in  examining  the  satellites  of  Jupiter  with 
a  telescope,  perceived  that  they  underwent  periodical  variations 
of  brightness.  These  variations  he  supposed  to  proceed  from  a 
rotation  of  the  satellites  upon  axes  which  caused  them  to  turn 
different  faces  towards  the  earth ;  and  from  repeated  and  careful 
observations  made  upon  them,  he  discovered  that  each  satellite 
made  one  turn  upon  its  axis  in  the  same  time  that  it  accomplished 


SATURN,    WITH    ITS   SATELLITES   AND   RING. 


223 


a  revolution  around  the  primary,  and  therefore,  like  the  moon, 
presented  continually  the  same  face  to  the  primary. 

SATURN,  WITH  ITS  SATELLITES  AND  RING-. 

379.  Saturn  shines  with  a  pnledull  light.    Its  apparent  diam- 
eter varies  less  than  6",  by  reason  of  the  change  of  distance, 
and  is  1C"  at  the  mean  distance.     The  eclipses  of  its  satellites 
indicate  that  it  is  opake,  and  illuminated  by  the  sun.     Saturn  is 
the  largest  of  the  planets,  next  to  Jupiter.     Its  equatorial  diame- 
ter is  9  times  that  of  the  earth,  or  72.000  miles;  and  its  volume 
is  670  times  that  of  the  earth.     The  rotation  on  jts  axis  is  per- 
formed in  lOh.  29rn.     The  inclination  of  its  axis  to  the  ecliptic 
is  about  62°.     Its  oblateness  is  -^. 

380.  Belts  of  Saturn.     The   disc  of  Saturn,  like  that  of 
Jupiter,  is  frequently  crossed  with  dark  bands,  or  belts,  in  a  di- 
rection parallel  to  its  equator.     But  Saturn's  belts  are  far  more 
indistinct  than  those  of  Jupiter.     Extensive  dusky  spots  are  also 
occasionally  seen  upon  its  surface.  (  See  Fig.  94.) 

The  cause  of  Saturn's  belts  is  doubtless  the  same  as  that  of  Jupiter's.  They 
accordingly  establish  the  existence  of  an  atmosphere  upon  the  surface  of  Saturn. 
The  results  of  Herschel's  observations  on  the  occupations  of  the  satellites  by 
the  planet,  indicate  the  existence  of  a  dense  atmosphere. 

3§1.  Saturn's  Ring.  The  planet  Saturn  is  distinguished 
from  all  the  other  planets  in  being  surrounded  by  a  broad,  thin, 
luminous  ring,  situated  in  the  plane  of  its  equator,  and  entirely 
detached  from  the  body  of  the  planet.  (See  Fig.  94.)  This 
ring  sometimes  casts  a  shadow  upon  the  planet,  and  is,  in  turn, 
at  times  partially  obscured  by  the  shadow  of  the  planet;  from 
which  we  conclude  that  it  is  opake,  and  receives  its  light  from 
the  sun. 

It  is  inclined  to  the 
plane  of  the  ecliptic  in 
an  angle  of  about  28°, 
and  during  the  motion 
of  Saturn  in  its  orbit  re- 
mains continually  paral- 
lel to  itself.  The  face  of 
the  ring  is,  therefore, 
never  viewed  perpendic- 
ularly from  the  earth, 
and  for  this  reason  nev- 
er appears  circular,  al 
though  such  is  its  actual 
form.  Its  apparent  form 
is  that  of  an  ellipse,  more 
or  less  eccentric,  accord-  FIG.  94. 

ing  to  the  obliquity  un- 
der which  it  is  viewed,  which  varies  with  the  position  of  Saturn 


224 


THE   PLANETS. 


in  its  orbit.  When  it  is  seen  under  the  larger  angles  of  obliqui- 
ty, it  appears  as  a  luminous  band  nearly  encircling  the  planet, 
and  is  visible  in  telescopes  of  small  power.  Stars  can  also  be 
seen  between  it  and  the  planet  in  these  positions.  At  other 
times,  when  viewed  very  obliquely,  it  can  be  seen  only  with  tele- 
scopes of  high  power.  When  it  is  approaching  the  latter  state, 
it  has  the  appearance  of  two  handles  or  ansce,  one  on  each  side 
of  the  planet. 

It  is  also  at  times  invisible.  This  is  the  case  whenever  the 
earth  and  sun  are  on  different  sides  of  the  plane  of  the  ring,  for 
the  reason  that  the  illuminated  face  is  then  turned  from  the  earth. 
When  the  plane  of  the  ring  passes  through  the  centre  of  the  sun, 
the  illuminated  edge  can  be  seen  only  in  telescopes  of  extraordi- 
nary power,  and  appears  as  a  thread  of  light  cutting  the  disc  of 
the  planet. 

3§2.  Circumstances  of  Disappearance  of  Ring.  Since 
the  orbit  of  Saturn  is  very  large  in  comparison  with  the  orbit  of 
the  earth,  the  plane  of  the  ring,  during  the  greater  part  of  the 
revolution  of  Saturn,  will  pass  without  the  orbit  of  the  earth; 
and  when  this  is  the  case  the  ring  will  be  visible,  as  the  earth 
and  sun  will  be  on  the  same  side  of  its  plane.  During  the  period, 
which  is  about  a  year,  that  the  plane  of  the  ring  is  passing  by 
the  orbit  of  the  earth,  the  earth  will  sometimes  be  on  the  sam<» 
side  of  it  as  the  sun,  and  sometimes  on  opposite  sides.  In  the 
latter  case  the  ring  will  be  invisible,  and  in  the  former  will  be 
seen  so  obliquely  as  to  be  visible  only  in  telescopes  of  consider- 
able or  great  power.  All  this  will  perhaps  be  better  understood 
on  consulting  Fig.  95,  where  efg  represents  the  orbit  of  the  earth. 


FIG.  95. 


The  appearances  of  the  ring  in  the  different  positions  of  the  planel 
in  its  orbit  are  delineated  in  the  figure. 

The  plane  of  the  ring  will  pass  through  the  sun  every  semi- 
revolution  of  Saturn,  or,  at  a  mean,  about  every  fifteen  years ; 
and  at  the  epochs  at  which  the  longitude  of  the  planet  is  re- 
spectively 170°  and  850°.  The  ring  will  then  disappear  once  in 


SATURN'S  RING.  225 

about  fifteen  years;  but,  owing  to  the  different  situations  of  the 
earth  in  its  orbit,  under  varied  circumstances:  and  the  disap- 
pearance will  occur  when  the  longitude  of  the  planet  is  about  170° 
or  350°.  The  ring  will  be  seen  to  the  greatest  advantage  when  the 
longitude  of  the  planet  is  not  far  from  80°,  or  260°.  The  last 
disappearance  took  place  in  1861;  the  next  will  be  in  187T. 
At  the  present  time  (1867)  the  north  face  of  the  ring  is 
visible. 

383.  Rotation  of  King.— Dimensions.  From  observa- 
tions made  upon  bright  spots  seen  on  the  face  of  the  ring,  Her- 
schel  discovered  that  it  rotated  from  west  to  east  about  an  axis 
perpendicular  to  its  plane,  and  passing  through  the  centre  of  the 
planet  (or  very  nearly).  The  period  of  its  rotation  is  lOh.  32m. 
It  is  remarkable  that  this  is  almost  the  exact  period  in  which  a 
satellite  assumed  to  be  at  a  mean  distance  equal  to  the  mean  dis- 
tance of  the  particles  of  the  ring,  would  revolve  around  the 
primary,  according  to  the  third  law  of  Kepler. 

The  breadth  of  the  ring  is  28,400  miles,  which  is  a  little  more 
than  one-half  greater  than  its  distance  from  the  surface  of  the 
planet,  and  exceeds  one-third  the  equatorial  diameter  of  the 
planet. 

3§4.  Divisions  of  the  Ring.  What  we  have  called  Sa- 
turn's Ring  consists  in  fact  of  two  principal  concentric  rings ; 
which  turn  together,  although  entirely  detached  from  each  other. 
The  void  space  between  them  is  perceived  in  telescopes  of  high 
power,  under  the  form  of  a  black  oval  line.  Calculations  from 
the  micrometric  measures  of  Professor  Struve  give  for  the 
breadth  of  the  inner  ring  16,500  miles,  and  of  the  outer,  10,150 
miles.  The  interval  between  the  rings  is  1,700  miles,  and  the 
distance  from  the  planet  to  the  inside  of  the  interior  ring  is 
18,300  miles.  The  thickness  of  the  rings  is  not  well  known ; 
the  edge  subtends  an  angle  less  than  3^",  which  at  the  distance 
of  the  planet,  answers  to  210  miles. 

The  division  of  the  ring  was  discovered  as  early  as  the  year  1665.  The  im- 
proved telescopes  in  the  hands  of  modern  observers,  have  revealed  the  existence 
of  a  dark  line  on  the  exterior  ring,  indicative  of  a  subdivision  of  this  ring.  It  is 
outside  of  the  middle  of  the  ring,  and  its  breadth  is  estimated  by  Dawes  at  about 
one-third  of  that  of  the  principal  division  of  the  whole  ring. 

385.  A  new  Ring  of  Saturn,  interior  to  the  other  two,  was  discovered  by  G.  P. 
Bond,  then  assistant  at  the  Observatory  at  Harvard  College,  on  the  llth  of  Novem- 
ber, 1850.  It  was  subsequently  observed  by  the  Messrs.  Bond  on  repeated  occasions 
from  that  date  to  the  7th  of  January,  1851.  It  shone  with  a  pale  dusky  light.  Its 
inner  edge  was  distinctly  defined,  but  the  side  next  the  old  ring  was  not  so  definite ; 
so  that  it  was  impossible  to  make  out  with  certainty  whether  the  new  was  con- 
nected with  the  old  ring  or  not. 

The  same  appearances  were  noticed  by  Dawes,  at  his  observatory  near  Maid- 
stone,  England,  on  the  25th  and  29th  of  November,  and  subsequently  by  Lassell, 
with  his  large  reflector,  at  Starfield,  near  Liverpool.  According  to  Dawes,  the 
breadth  of  the  new  ring  is  1".7.  or  7,200  miles;  and  its  distance  from  the  inner 
edge  of  the  bright  ring  0".3,  or  1,270  miles. 

386.  Form  of  Cross  Section  of  the  Ping.  Bessel  has  shown  that  the  double 
nng  is  not  bounded  by  parallel  plane  surfaces.  He  infers  this  to  be  the  case  from 

15 


226  THE  PLANETS. 

the  fact  that  at  almost  every  disappearance  or  reappearance  of  the  ring,  the  two 
ansse  have  not  disappeared  or  reappeared  at  the  same  time.  He  has  also  found, 
from  a  discussion  of  the  observations  which  have  been  made  upon  the  disappear- 
ances and  reappearances  of  the  ring,  that  they  cannot  be  satisfied  by  supposing 
the  two  faces  of  the  ring  to  be  parallel  planes.  In  view  of  all  the  facts,  it  seems 
most  probable  that  the  cross  section  of  each  ring  has  the  approximate  form  of  a 
very  eccentric  ellipse  instead  of  a  rectangle,  and  that  it  varies  somewhat  in  size 
from  one  part  of  the  ring  to  another.  It  may  have  irregularities  on  its  surface,  as 
great  or  greater  than  those  which  diversify  the  surface  of  the  earth. 

387.  Centre  of  Gravity  of  each  Ring — Stability  of  the  Rings.    Whatever  may 
be  the  form  of  the  rings,  their  matter  is  not  uniformly  distributed;   for  microme- 
tric  measurements  of  great  delicacy  made  by  Struve,  have  made  known  the  fact, 
that  the  rings  are  not  concentric  with  the  planet,  but  that  their  centre  of  gravity 
revolves  in  a  minute  orbit  about  the  centre  of  the  planet.    Laplace  had  previously 
inferred,  from  the  principle  of  gravitation,  that  this  circumstance  was  essential  to 
the  stability  of  the  rings.     He  demonstrated  that  if  the  centre  of  gravity  of  either 
ring  were  once  strictly  coincident  with  the  centre  of  gravity  of  the  planet,  the 
slightest  disturbing  force,  such  as  the  attraction  of  a  satellite,  would  destroy  the 
equilibrium  of  the  ring,  and  eventually  cause  the  ring  to  precipitate  itself  upon  the 
planet. 

388.  Physical  Constitution  of  the  Ring.     G.  P.  Bond  has  propounded  a  bold 
and  ingenious  theory,  relative  to  the  physical  constitution  of  Saturn's  rings ;  which 
is,  that  "  they  are  in  a  fluid  state,  and  within  certain  limits  change  their  form  and 
position  in  obedience  to  the  laws  of  equilibrium  of  rotating  bodies."    He  conceives 
also,  that  under  peculiar  circumstances  of  disturbance  several  subdivisions  of  the 
two  fluid  rings  may  take  place,  and  continue  for  a  short  time  until  the  sources  of 
disturbance  are  removed,  when  the  parts  thrown  off  would  again  reunite.    Profes- 
for  Pierce  has  followed  up  the  speculations  of  Bond,  by  undertaking  to  demon- 
strate from  purely  mechanical  considerations,  that  Saturn's  ring  cannot  be  solid. 
He  maintains  that  there  is  no  conceivable  form  of  irregularity,  and  no  combination 
of  irregularities,  consistent  with  an  actual  ring,  which  would  serve  to  retain  it  per- 
manently about  the  primary  if  it  were  solid.     He  is  led  by  his  investigations  to 
the   curious  result,  that  Saturn's  ring  is  sustained  in  a  position  of  stable  equili- 
brium about  the  planet,  solely  by  the  attractive  power  of  his  satellites ;  and  that 
no  planet  can  have  a  ring  unless  it  is  surrounded  by  a  sufficient  number  of  proper- 
ly arranged  satellites.    Upon  the  theory  of  the  development  of  heat  by  collision  with 
the  ether  of  space  (377),  the  temperature  of  the  mass  of  Saturn's  rings  should  be 
much  higher  than  that  of  the  body  of  the  planet,  as  its  actual  velocity  of  rotation 
is  nearly  twice  as  great,  and  the  possibility  of  a  liquid  condition  of  its  mass  may 
be  admitted. 

389.  Origin.    In  respect  to  the  origin  of  Saturn's  ring,  Sir  John  Herschel  has 
offered  the  interesting  suggestion,  that  as  the  smallest  difference  of  velocity  in 
space  between  the  planet  and  ring  must  infallibly  precipitate  the  latter  on  the 
former,  never  more  to  separate,  it  follows  either  that  their  motions  in  their  common 
orbit  around  the  sun  must  have  been  adjusted  by  an  external  power  with  the 
minutest  precision,  or  that  the  ring  must  have  been  formed  about  the  planet  while 
subject  to  their  common  orbital  motion,  and  under  the  full  influence  of  all  the  act- 
ing forces.     The  latter  supposition  accords  with  Laplace's  theory  of  the  progres- 
sive development  of  the  planetary  system, 

390.  The  Satellites  of  Saturn  were  discovered,  the  6th,  in 
the  order  of  distance,  by  Hu yghens,  in  1655,  with  a  telescope  of 
12  feet  focus;  the  3d,  4th,  5th  and  8th,  by  Dominique  Cassini, 
between  the  years  1670  and  1685,  with  refracting  telescopes  of 
100  and  136  feet  in  length ;  and  the  1st  and  2d,  by  Sir  William 
Herschel,  in  1789,  with  his  great  reflecting  telescope  of  40  feet 
focus.     All  these  but  the  1st  and  2d,  are  visible  in  a  telescope  of 
large  aperture,  with   a  magnifying   power  of  200.     The   7th 
satellite,    in    the    order   of   distance    from    the   primary,    was 
discovered  by  the  Messrs.  Bond,    with   the  great  refractor  of 


NEPTUNE.  227 

the  Cambridge  Observatory,  on  the  16th  of  September,  1848  ; 
and  observed  two  days  afterwards  by  Lassell.  It  has  received 
the  name  of  Hyperion.  The  periods  of  revolution  and  the  mean 
distances  of  the  satellites  of  Saturn  from  their  primary,  together 
with  the  mythological  names  proposed  for  them  by  Sir  John 
Herschel,  are  given  in  Table  VI.  I 

All  of  Saturn's  satellites,  with  the  exception  of  the  8th,  re- 
volve very  nearly  in  the  plane  of  the  ring,  and  of  the  equator 
of  the  primary.  The  orbit  of  the  8th  is  inclined  under  a  con- 
siderable angle  to  this  plane.  The  6th  satellite  is  much  the  larg- 
est, and  is  estimated  to  be  not  much  inferior  to  Mars  in  size. 
The  others  interior  to  this,  diminish  in  size,  towards  the  ring. 
The  1st  and  2d  are  so  small,  and  so  near  the  ring,  that  they  have 
never  been  discerned  but  with  the  most  powerful  telescopes 
which  have  yet  been  constructed,  and  with  these  only  at  the  time 
of  the  disappearance  of  the  ring  (to  ordinary  telescopes),  when 
they  have  been  seen  as  minute  points  of  light  skirting  the  narrow 
line  of  the  luminous  edge  of  the  ring.  The  new  satellite  (the 
7th)  is  described  as  fainter  than  either  of  these  two  interior 
satellites,  discovered  by  Sir  William  Herschel. 

The  8th  satellite  is  subject  to  periodical  variations  of  lustre, 
which  indicate  a  rotation  about  an  axis  in  the  period  of  a  sidereal 
revolution  of  Saturn. 


URANUS  AND  ITS  SATELLITES. 

891.  The  planet  now  known  by  the  name  of  Uranus,  was  dis- 
covered by  Sir  William  Herschel.  It  is  not  visible  to  the  naked 
eye,  except  in  opposition,  when  it  becomes  barely  discernible. 
In  a  telescope  it  appears  as  a  small,  round,  uniformly  illuminated 
disc.  Its  apparent  diameter  is  about  4",  from  which  it  never 
varies  much,  owing  to  the  small  size  of  the  earth's  orbit  in  com- 
parison with  its  own.  Its  real  diameter  is  33,000  miles,  and  its 
bulk  73  times  that  of  the  earth.  Analogy  leads  us  to  believe 
that  this  planet  is  opake  and  turns  on  an  axis,  but  there  is  no 
positive  evidence  that  this  is  the  case. 

Of  the  eight  satellites  of  Uranus,  six  were  discovered  by  Her- 
schel, one  by  Lassell,  and  one  by  0.  Struve. 


NEPTUNE. 

392.  It  is  a  remarkable  fact  that  the  existence  of  this  planet 
was  first  detected  from  the  disturbances  it  produced  in  the  motions 
of  Uranus.  It  having  been  ascertained  that  there  were  out- 
standing inequalities  in  the  motion  of  this  planet,  which  could 
not  be  referred  to  the  action  of  the  other  planets,  Le  Verrier,  the 


2:?S  THE  PLANETS. 

eminent  French  astronomer,  undertook  in  1845  the  problem  of 
determining  the  orbit  and  mass  of  a  planet  capable  of  producing 
such  inequalities.  The  same  problem  was  independently  under- 
taken and  successfully  solved  by  Mr.  Adams,  of  Cambridge, 
England.  Le  Yerrier,  as  the  final  result  of  his  computations, 
indicated  the  probable  place  of  the  theoretical  planet  in  the 
heavens ;  and  Dr.  Galle,  of  Berlin,  upon  directing  the  great  tele- 
scope of  the  Eoyal  Observatory  on  the  region  indicated,  on  the 
evening  of  the  23d  of  September,  1846,  descried  the  new  planet 
within  1°  of  its  most  probable  place,  as  assigned  by  Le  Yerrier. 

The  apparent  diameter  of  Neptune  is  a  little  less  than  3".  Its 
real  diameter  is  36,000  miles;  and  its  volume  93  times  that  of 
the  earth.  Neptune,  like  Uranus,  is  destitute  of  visible  spots 
and  belts,  and  the  period  of  its  axial  rotation  is  unknown. 

Neptune's  satellite  was  discovered  by  Lassell  in  1846.  The 
same  observer  has  since  obtained  traces  of  the  existence  of  a 
second  satellite. 

THE  PLANETOIDS. 

393.  Yesta  is  the  brightest  of  the  minor  planets.  In  the  tele- 
scope, it  appears  as  a  star  of  6th  or  7th  magnitude.  Pallas, 
Ceres,  and  Juno  appear  of  the  7th  or  8th  magnitude.  The  great 
majority  of  the  other  planetoids  are  of  the  10th  or  llth  magni- 
tude. Pallas  is  the  largest  of  this  class  of  bodies.  According  to 
Lamont,  Director  of  the  Koyal  Observatory,  Munich,  its  diame- 
ter is  670  miles.  The  diameter  of  Yesta  is  believed  not  to  exceed 
300  miles ;  and  that  of  Ceres  to  be  somewhat  smaller.  Juno  is 
the  smallest  of  the  four  planetoids  first  discovered.  All  of  the 
other  minor  planets  are  supposed  to  be  less  than  100  miles  in 
diameter. 


COMETS. 


229 


CHAPTER  XVIH. 

COMETS. 

THEIR  GENERAL  APPEARANCE:— VARIETIES  OP 
APPEARANCE. 

394.  The  general  appearance  of  comets  is  that  of  a  mass  of 
some  luminous  nebulous  substance,  to  which  the  name  Coma  has 
been  given,  condensed  towards  its  centre  around  a  brilliant 
nucleus  that  is  in  most  cases  not  very  distinctly  defined ;  from 
which  proceeds,  in  a  direction  opposite  to  the  sun,  a  stream  of 


FIG.  96. 

similar  but  less  luminous  matter,  called  the  Tail  or  Train  of  the 
comet  (Fig.  96).  The  nucleus,  with  the  surrounding  coma,  forms 
the  Head  of  the  comet. 

The  tail  gradually  increases  in  width,  and  at  the  same  time 
diminishes  in  distinctness  from  the  head  to  its  extremity,  where 
it  is  generally  many  times  wider  than  at  the  head,  and  fades 
away  until  it  is  lost  in  the  general  light  of  the  sky.  It  is,  in 


230  COMETS. 

general,  less  bright  alon^  its  middle  than  at  the  borders.  From 
this  cause  the  tail  sometimes  seems  to  be  divided,  along  a  greater 
or  less  portion  of  its  length,  into  two  separate 
tails  or  streams  of  light,  with  a  comparative  dark 
space  between  them.  Ordinarily  it  is  not  straight, 
that  is,  coincident  with  a  great  circle  of  the 
heavens,  but  concave  towards  that  part  of  the 
heavens  which  the  comet  has  just  left.  This 
curvature  of  the  tail  is  most  observable  near  its 
extremity.  The  most  remarkable  example  is 
that  of  the  comet  of  1744,  which  was  bent  so  as 
to  form  nearly  a  quarter  of  a  circle.  Nor  does 
the  general  direction  of  the  tail  usually  coincide 
exactly  with  the  great  circle  passing  through  the 
suii  and  the  head  of  the  comet,  but  deviates 
more  or  less  from  this,  the  position  of  exact  op- 
position to  the  sun  in  the  heavens,  on  the  side 
towards  the  quarter  of  the  heavens  just  traversed 
by  the  comet.  This  deviation  is  quite  different 
for  different  comets,  and  varies  materially  for 
the  same  comet  while  it  continues  visible.  It 
has  even  amounted  in  some  instances  to  a  right 
angle. 

395.  Variation*  of  Length  of  Tail.  The 
apparent  length  of  the  tail  varies,  from  one  comet 
to  another,  from  zero  to  100°  and  more;  and 
ordinarily  the  tail  of  the  same  comet  increases 
and  diminishes  very  much  in  length  during  the 
period  of  its  visibility.  When  a  comet  first  ap- 
pears, in  general  no  tail  is  perceptible,  and  its 
light  is  very  faint.  As  it  approaches  the  sun,  it 
becomes  brighter;  the  tail  also,  after  a  time, 
shoots  out  from  the  coma,  and  increases  from 
day  to  day  in  extent  and  distinctness.  As  the 
comet  recedes  from  the  sun,  the  train  precedes  the 
head,  being  still  on  the  opposite  side  from  the 
sun,  and  grows  less  and  less  at  the  same  time 
that,  along  with  the  head,  it  decreases  in  bright- 
ness, till  at  length  the  comet  resumes  nearly  its 
first  appearance,  and  finally  disappears.  (See 
Fig.  98.)  It  sometimes  happens  that,  owing  to 
FIG.  97.  peculiar  circumstances,  a  cornet  does  not  make 

Great  Comet  c/1843  ^s  appearance  in  the  firmament  until  after  it  has 
passed  the  sun  in  the  heavens,  and  not  until  it 
has  attained  to  more  or  less  distinctness,  and  is  furnished  with  a 
train  of  considerable  or  even  great  length.  This  was  remarkably 
the  case  with  the  great  comet  of  1843.  (See  Art.  237;  also 
Fig.  97.) 


THEIR  GENERAL  APPEARANCE.  231 

396.  Effect*  of  the  Position  of  the  Earth,  on  the  appar- 
ent size  and  brightness  of  a  comet.  The  tail  of  a  comet  is  the  longest, 
and  the  whole  comet  is  intrinsically  the  most  luminous,  not  long 
after  it  has  passed  its  perihelion.  Its  apparent  size  and  lustre  will 
not,  however,  necessarily  be  the  greatest  at  this  time,  as  they  will 
depend  upon  the  distance  and  position  of  the  earth,  as  well  as  the 
actual  size  and  intrinsic  brightness  of  the  comet.  To  illustrate 


FIG.  98. 

this,  let  abed  (Fig.  98)  represent  the  orbit  of  the  earth,  and  MPN 
the  orbit  of  a  comet,  having  its  perihelion  at  P.  Now,  if  the 
earth  should  chance  to  be  at  a  when  the  comet,  moving  towards 
its  perihelion,  is  at  r,  it  might  very  well  happen  that  the  comet 
would  appear  larger  and  more  distinct  than  when  it  bad  reached 
the  more  remote  point  s,  although  when  at  the  latter  point  it 
would  in  reality  be  larger  and  brighter  than  when  at  r.  It 
would  be  the  most  conspicuous  possible  if  the  earth  should  be  in 
the  vicinity  of  c  or  b  soon  after  the  perihelion  passage ;  and  it 
would  be  the  least  conspicuous  possible  if  the  comet  be  supposed 
to  be  moving  in  the  direction  NPM,  and  to  pass  from  N  around 
to  M,  while  the  earth  is  moving  around  from  a  to  b  or  c  ;  so  as  to 
be  continually  comparatively  remote  from  the  comet,  and  so  that 
the  comet  will  be  in  conjunction  with  the  sun  at  the  time  after 
the  perihelion  passage  when  its  actual  size  and  intrinsic  lustre 
are  the  greatest.  It  is  to  be  observed  that  the  apparent  lustre  of 
^  comet  is  sometimes  very  much  enhanced  by  the  great  obli- 
quity of  the  tail,  in  some  of  its  positions,  to  the  line  of  sight. 
This  seems  to  have  been  the  case  with  the  comet  of  1843,  on 
February  28  (see  Fig.  63),  and,  it  has  been  already  intimated, 


232  COMETS. 

was  one  reason  of  its  being  so  very  bright  as  to  be  seen  in  open 
day  in  the  immediate  vicinity  of  the  sun. 

Since  the  earth  may  have  every  variety  of  position  in  its  orbit 
at  the  successive  returns  of  the  same  comet  to  its  perihelion,  it  will 
be  seen,  on  examining  Fig.  98,  that  the  circumstances  of  the  ap- 
pearance and  disappearance  of  the  comet,  as  well  as  its  size  and 
distinctness,  may  be  very  different  at  its  different  returns.  This 
has  been  strikingly  true  in  the  case  of  Halley's  Comet.  Biela's 
Comet  was  also  invisible  on  its  return  to  its  perihelion  in  1839,  by 
reason  of  its  continual  proximity  to  the  line  of  direction  of  the 
sun  as  seen  from  the  earth,  and  its  great  distance  from  the  earth. 

397.  Varieties  of  Aspect.  Individual  comets  offer  consider- 
able varieties  of  aspect.  Some  comets  have  been  seen  which 
were  wholly  destitute  of  a  tail:  such,  among  others,  was  the 
comet  of  1 682,  which  Cassini  describes  as  being  as  round  and  as 
bright  as  Jupiter.  Others  have  had  more  than  one  luminous 
train.  The  cornet  of  1744  was  provided  with  six,  which  were 
spread  out  like  an  immense  fan,  through  an  angle  of  117° ;  and 
that  of  1823  with  two,  one  directed  from  the  sun  in  the  heavens, 
and,  what  is  very  remarkable,  another  smaller  and  fainter  one 
directed  towards  the  sun.  Others  still  have  had  no  perceptible 
nucleus,  as  the  cornets  of  1795  and  1804. 

The  comets  that  are  visible  only  in  telescopes,  which  are  very 
numerous,  have  generally  no  distinct  nucleus,  and  are  often  en- 
tirely destitute  of  every  vestige  of  a  tail.  They  have  the  appear- 
ance of  round  masses  of  luminous  vapor,  somewhat  more  dense 
towards  the  centre.  Such  are  Encke's  and  Biela's  comets.  (Fig. 
99.)  The  point  of  greatest  condensation  is  often  more  or  less 


FIG.  99. — Encke's  Cornel, 


removed  from  the  centre  of  figure  on  the  side  towards  the  sun  ; 
and  sometimes  also  on  the  opposite  side. 

«9§.  The  Comets  which  have  had  the  Longest  Trains, 

are  those  of  1630,  1769,  and  1618.     The  tail  of  the  great  comet 


FORM  AND  STRUCTURE  OF  COMETS.  233 

of  1680,  when  apparently  the  longest,  extended  to  a  distance  of 
70°  from  the  head;  that  of  the  comet  of  1769,  a  distance  of  97°: 
and  that  of  the  comet  of  1618,  104°.  These  are  the  apparent 
lengths  as  seen  at  certain  places.  By  reason  of  the  different  de- 
grees of  purity  and  density  of  the  air  through  which  it  is  seen, 
the  tail  of  the  same  comet  often  appears  of  a  very  different  length 
to  observers  at  different  places.  Thus,  the  comet  of  1769,  which 
at  the  Isle  of  Bourbon  seemed  to  have  a  tail  of  97°  in  length,  at 
Paris  was  seen  with  a  tail  of  only  60°.  From  this  general  fact 
we  may  infer  that  the  actual  train  extends  an  unknown  distance 
beyond  the  extremity  of  the  apparent  train. 


FORM.  STRUCTURE,  AND  DIMENSIONS  OF  COMETS. 

399.  The  general  form  and  structure  of  comets,  so  far  as  they 
can  be  ascertained  from  the  study  of  the  details  of  their  appear- 
ance, may  be  described  as  follows :  The  head  of  a  comet  consists 
of  a  central  nucleus,  or  mass  of  matter  brighter  and  denser  than 
the  other  portions  of  the  comet,  enveloped  on  the  side  towards 
the  sun,  and  ordinarily  at  a  great  distance  from  its  surface  in 
comparison  with  its  own  dimensions,  by  a   globular  nebulous 
mass  of  great  thickness,  called  the  Nebulosity,  or  nebulous  En- 
velope.    This,  it  is  said,  never  completely  surrounds  the  nucleus, 
except  in  the  case  of  comets  which  have  no  tails.     It  forms  a 
sort  of  hemispherical  cap  to  the  nucleus  on  the  side  towards  the 
sun.     Its  form,  however,  is  not  truly  spherical,  but  varies  be- 
tween this  and  that  of  a  paraboloid  having  the  nucleus  in  its 
focus  and  its  vertex  turned  towards  the  sun.     The  tail  begins 
where  the  nebulosity  terminates,  and  seems,  in  general,  to  be 
merely  the  continuation  of  this  in  nearly  a  straight  line  beyond 
the  nucleus.     There  is  ordinarily,  as  has  been  already  intimated, 
a  distinct  space  containing  less  luminous  matter  between  the 
nucleus  and  the   nebulosity,  but  this  is  not  always  the  case. 
The  tail  of  a  comet  has  the  shape   of  a  hollow  truncated  cone, 
with  its  smaller  base  in  the  nebulosity  of  the  head ;  with  this 
difference,  however,  that  the  sides  are  usually  more  or  less  curved. 
That  the  tail  is  hollow  is  evident  from  the  "fact,  already  noticed, 
that  on  whichever  side  it  is  viewed  it  appears  less  bright  along 
the  middle  than  at  the  borders.     There  can  be  less  luminous 
matter  on  a  line  of  sight  passing  through  the  middle,  than  on 
one  passing  near  one  of  the  edges,  only  on  the  supposition  that 
the  tail  is  hollow.     The  whole  tail  is  generally  bent  so  as  to  be 
concave  towards  the  regions  of  space  which  the  comet  has  just 
left. 

400.  Multiple  Tails.     In  some  instances  the  nucleus  is  fur- 
nished with  several  envelopes  concentric  with  it :    which  are 
formed  in  succession  as  the  comet  approaches  the  sun,  and  then 


234:  COMET3. 

recedes  from  it  again.  For  example,  the  comet  of  1744,  eight 
days  after  the  perihelion  passage,  had  three  envelopes.  Some- 
times each  of  them  is  provided  with  a  tail.  Each  of  these  sev- 
eral tails  being  hollow,  may  in  consequence  appear  so  faint  along 
its  middle  as  to  have  the  aspect  of  two  distinct  tails.  A  comet 
which  has  in  reality  three  separate  trains,  might  thus  appear  to 
be  supplied  with  six,  as  was  the  cornet  of  1744.  If  the  different 
envelopes  were  not  distinctly  separate  from  each  other,  then  all 
the  trains  would  appear  to  proceed  from  the  same  nebulous 
mass. 

Supernumerary  tails,  shorter  and  less  distinct  than  the  •  princi- 
pal one,  are  by  no  means  uncommon ;  but  they  generally  appear 
quite  suddenly,  and  as  suddenly  disappear  in  a  few  days,  as  if 
the  stock  of  materials  from  which  they  were  supplied  had  be- 
come exhausted. 

4O1.  The  general  Position  of  the  Tail  of  a  Comet  is 
nearly  but  not  exactly  in  the  prolongation  of  the  line  of  the 
centres  of  the  sun  and  head,  or  of  the  radius- vector  of  the  comet. 
(See  Fig.  98.)  It  deviates  from  this  line  on  the  side  of  the  re- 
gions of  space  which  the  comet  has  just  left ;  and  the  angle  of 
deviation,  which,  when  the  comet  is  first  seen  at  a  distance  from 
the  sun,  is  very  small  or  not  at  all  perceptible,  increases  as  the 
comet  approaches  the  sun,  and  attains  to  its  maximum  value 
soon  after  the  perihelion  passage ;  after  which  it  decreases,  and 
finally,  at  a  distance  from  the  sun,  becomes  insensible.  For  ex- 
ample, the  angle  of  deviation  of  the  tail  of  the  great  comet  of 
1811  attained  to  its  maximum  about  ten  days  after  the  perihelion 
passage,  and  was  then  about  11°*  In  the  case  of  the  comet  of 
1664,  the  same  angle  about  two  weeks  after  the  perihelion  pas- 
sage was  43°,  and  was  then  decreasing  at  the  rate  of  8°  per  day. 

The  comet  of  1 823  might  seem  to  present  an  exception  to  the 
general  fact  that  the  tail  of  a  comet  is  nearly  opposite  to  the 
sun ;  but  Arago  has  suggested  that  the  probable  cause  of  the 
singular  phenomenon  of  a  secondary  tail,  apparently  directed 
towards  the  sun  in  the  heavens,  was  that  the  earth  was  in  such 
a  position  that  the  two  tails,  although  in  fact  inclined  to  each 
other  under  a  small  angle,  were  directed  towards  different  sides 
of  the  earth,  and  thus  were  referred  to  the  heavens  so  as  to  ap- 
pear nearly  opposite. 

The  same  principle  will  serve  to  show  that  the  deviation  of  the 
train  of  a  cornet  from  the  position  of  exact  opposition  to  the 
sun  may  appear  to  be  much  greater  than  it  actually  is,  by  reason 
of  the  earth's  happening  to  be  within  the  angle  formed  by  the 
direction  of  the  train  with  the  radius-vector  prolonged. 

4O2.  Vast  Size  of  Comets.  Comets  are  the  most  volumi- 
nous bodies  in  the  solar  system.  The  tail  of  the  great  comet  of 
1680  was  found  by  Newton  to  have  been,  when  longest,  no  less 
than  123,000,000  miles  in  length.  The  remarkable  comet  of 


DIMENSIONS  OF  COMETS.  235 

1843,  about  three  weeks  after  its  perihelion  passage,  had  a  tail 
of  over  108,000,000  miles  in  length.  Other  comets  have  had 
trains  from  fifty  to  one  hundred  million  miles  long.  The  heads 
of  cornets  are  generally  tens,  and  often  hundreds,  of  thousands 
of  miles  in  diameter.  That  of  the  great  comet  of  1811  had  a 
diameter  of  over  1,000,000  miles ;  that  of  Halley's  comet,  in 
1836,  a  diameter  of  350,000  miles,  and  that  of  Eneke's  comet, 
in  1828,  a  diameter  of  over  300,000  miles.  The  head  of  the 
great  comet  of  1843  was  about  30,000  miles  in  diameter. 

403.  The  Nuclei  of  comets,  so  far  as  they  have  been  accu- 
rately determined,  do  not  exceed  a  few  hundred  miles  in  diame- 
ter.    For  example,  the  great  comet  of  1811  had  a  nucleus  of 
428  miles,  and  that  of  1798  one  of  125  miles  in  diameter.     In- 
stances are  cited  of  comets  with  nuclei  of  several  thousand  miles 
in  diameter  (e.  g.,  the  third  comet  of  1845,  and  the  fourth  comet 
of  1825) ;  but  there  is  little  reason  to  doubt  that  in  these  cases, 
the  apparent  telescopic  nucleus  ordinarily  observed  was  measured, 
instead  of  the  true  nucleus,  which  is  only  occasionally  seen. 
AVhen  a  comet  is  viewed  with  the  naked  eye,  it  usually  offers 
the  appearance  of  a  star-like  nucleus  at  the  centre  of  the  head. 
Telescopes  resolve  this,   more  or  less,   into  a  bright  nebulous 
mass,  which  is  the  ordinary  telescopic  nucleus.     But  occasionally 
they  show,  in  the  case  of  a  bright  comet,  within  this  a  stellar 
point,  distinguished  by  its  brightness  and  appearance  of  solidity 
from  the  nebulosity  about  it.     This  is  the  true  nucleus.     The 
nucleus,  so-called,  of  Donati's  comet,  is  stated  to  have  been  5.600 
miles  in  diameter,  but  according  to  Bond,  the  true  nucleus  that 
was  occasionally  discernible  in  his  telescope,  was  less  than  500 
miles  in  diameter. 

404.  Variation  of  Dimensions.  The  dimensions  of  comets 
are  subject  to  continual  variations.     The  tail  increases  in  actual 
length  as  the  comet  approaches  the  sun,  and  attains  its  greatest 
size  a  certain  time  after  the  perihelion  passage ;  After  which  it 
gradually  decreases.      The   head,  on   the   contrary,  diminishes 
in  size  during  the  approach  to  the  sun,  and  augments  during  the 
recess  from  him.     These  changes  of  dimension,  both  in  the  case 
of  the  head  and  of  the  tail  of  the  comet,  are  often  very  great, 
and  sometimes  quite  sudden  and  rapid.     Encke's  comet,  at  its 
return  in  1828,  in  the  course  of  two  months,  while  its  distance 
from  the  sun  was  diminished  in  the  ratio  of  1  to  3,  underwent 
an  apparent  diminution  of  volume  in  the  ratio  of  16,000  to  1. 
The  apparent  nucleus  of  Donati's  comet  was  1,000  times  less  soon 
after  the  perihelion  passage  than  when  it  was  previously  seen  at 
a  distance  two  or  three  times  greater.     The  tail  of  the  great 
2omet  of  1843  increased  in  length  after  the  perihelion  passage,  at 
the  rate  of  5,000,000  miles  per  day  ;  and  that  of  Donati's  comet 
increased  in  length  for  ten  days  after  the  perihelion  passage,  at 
the  average  rate  of  2,500,000  miles  per  day. 


236  COMETS. 


PHYSICAL  CONSTITUTION  OF  COMETS. 

405.  Small  Ulass  and  Density.     The   quantity    of  matter 
which  enters  into  the  constitution  of  a  comet  is  exceedingly 
small.     This  is  proved  by  the  fact  that  comets  have  had  no  influ- 
ence upon  the  motions  of  the  planets  or  satellites,  although  they 
have,  in  many  instances,  passed  near  these  bodies.     The  comet 
of  1770,  which  was  quite  large  and  bright,  passed  in  close  prox- 
imity to  Jupiter's  satellites,  without  deranging  their  motions  in 
the  least  perceptible  degree.     Moreover,  since  this  small  quan- 
tity of  matter  is  dispersed  over  a  space  of  tens  of  thousands  or 
millions  of  miles  (if  we  include  the  tail),  in  linear  extent,  the 
nebulous  matter  of  comets  must  be  incalculably  less  dense  than 
the  solid  matter  of  the  planets.     In  fact,  the  cometic  matter, 
with  the  exception  perhaps  of  that  of  the  nucleus,  is  inconceiva- 
bly more  rare  and  subtile  than  the  lightest  known  gas,  or  the 
most  evanescent  film  of  vapor  that  ever  makes  its  appearance  in 
our  sky  ;  for  faint  telescopic  stars  are  distinctly  visible  through 
all  parts  of  the  comet,  with,  it  may  be,  the  exception  of  the 
nucleus,  notwithstanding  the  great  space  occupied  by  the  matter 
of  the  comet,  which  the  light  of  the  star  traverses.     The  matter 
of  the  tail  of  a  comet  is  even  more  attenuated  than  that  of  the 
general  mass  of  the  nebulosity  of  the  head,  but  is  apparently  of 
the  same  nature,  and  derived  from  the  head. 

406.  Nucleus  and  Nebulosity.     The  nucleus  is  supposed 
by  some  astronomers  to  be,  in  some  instances,  a  solid,  partially 
or  wholly  convertible  into  vapor  under  the  influence  of  the  sun ; 
by  others,  to  be  in  all  cases  the  same  species  of  matter  as  the 
nebulosity,   only  in   a   more  condensed  state;    and   by  others 
still,  to  be  a  solid  of  permanent  dimensions,  with  a  thick  stratum 
of  condensed  vapors  resting  upon  its  surface.     Whichever  of 
these   views   be   adopted,   it   is   a   matter   of  observation   that 
the   nebulosity   frequently    receives    fresh    supplies    of   matter 
from  the  nucleus.     It  was  the  opinion  of  Sir-William  Herschel, 
and  it  lias  been  the  more  generally  received   notion  since  his 
time,  that  the  nucleus  of  a  comet  is  surrounded  with  a  transpar- 
ent atmosphere  of  vast  extent,  within  which  the  nebulous  enve- 
lope floats,  as  do  clouds  in  the  earth's  atmosphere.     But  Olbers, 
and  after  him  Bessel,  conceives  the  nebulous  matter  of  the  head 
to  be  either  in  the  act  of  flowing  away  into  the  tail  under  the 
influence  of  a  repulsion  from  the  nucleus  and  the  sun,  or  in  a 
state  of  equilibrium  under  the  action  of  these  forces  and  the 
attraction  of  the  nucleus. 

407.  Luminosity  of  Comets.  Observations  with  the  polari- 
scope  have  established  that  comets  shine  in  a  great  degree  by 
reflected  light  This  is  especially  true  of  the  tail  of  the  comet ;  the 
nucleus  and  nebulosity  present  feeble  traces  of  polarization,  and, 


FORMATION  OF  THE  TAILS  OF  COMETS.  237 

we  mast  therefore  conclude,  emit  a  strong  light  of  their  own,  or 
shine  wholly  by  light  radiantly  reflected.  If  the  head  of  a  comet 
shone  entirely  by  reflected  light,  and  the  amount  of  reflecting 
surface  remained  constantly  the  same,  its  apparent  brightness 
would  be  inversely  proportional  to  the  product  of  the  squares 
of  the  distances  from  the  sun  and  earth.  By  this  rule,  the  head 
of  Donati's  comet  should  have  been  188  times  brighter  on  the 
2d  of  October  than  on  June  loth ;  whereas  it  was  actually  6,300 
times  brighter.  From  which  we  may  infer  that  the  quantity  of 
light  emanating  from  it  bad  increased  in  the  proportion  of  33  to 
1.  This  increase  of  actual  light  was  confined  chiefly  to  the 
nebulosity  of  the  head,  and  is  probably  attributable,  in  a  great 
degree,  to  an  augmentation  of  the  quantity  of  nebulous  matter 
received  from  the  nucleus. 


CONSTITUTION  AND  MODE  OF  FORMATION  OF  THE  TAILS 
OF  COMETS. 

4O§.  Upon  this  topic  we  may  lay  down  the  following  postu- 
lates: 1.  The  general  situation  of  the  tail  of  a  comet  with  respect 
to  the  sun,  shows  that  the  sun  is  concerned,  either  directly  or 
indirectly,  in  its  formation.  The  changes  which  take  place  in 
the  dimensions  of  a  comet,  both  in  approaching  the  sun  and 
receding  from  it,  conduct  to  the  same  inference.  2.  Since  the 
tail  lies  in  the  direction  of  the  radius- vector  prolonged  beyond 
the  head,  the  particles  of  matter  of  which  it  is  made  up  must 
have  been  driven  off  by  some  force  exerted  in  a  direction  from 
the  sun.  3.  This  force  cannot  emanate  from  the  nucleus,  for 
such  a  force  would  expel  the  nebulous  matter  surrounding  the 
nucleus  in  all  directions,  instead  of  one  direction  only.  It  is, 
however,  conceivable  that,  as  Olbers  supposes,  the  nebulous 
matter  is,  in  the  first  instance,  expelled  from  the  nucleus  by  its 
repulsive  action,  taking  effect  chiefly  on  the  side  towards  the 
sun,  and  afterwards  driven  past  the  nucleus  into  the  tail  by  a 
repulsion  from  the  sun.  4.  There  seems,  then,  to  be  little  room 
to  doubt  that  the  matter  of  the  tail  is  driven  from  the  head  by 
some  force  foreign  to  the  comet,  and  taking  effect  from  the  sun 
outwards.  5.  This  force,  whatever  may  be  its  nature,  extends 
far  beyond  the  earth's  orbit ;  for  comets  have  been  seen  provided 
with  tails  of  great  length,  though  their  perihelion  distance  ex- 
ceeded the  radius  of  the  earth's  orbit  (e.g.,  the  great  comet  of 
1811).  6.  It  is  natural  to  suppose  that,  like  all  central  forces, 
the  repulsive  force  exerted  by  the  sun  upon  cometic  matter  varies  in- 
versely as  the  square  of  the  distance.  This  law  of  variation  has  in 
fact  been  established  by  the  investigations  of  Bessel  and  Profes- 
sor Pierce,  and  confirmed  by  the  author's  determination  of  the 
form  and  dimensions  of  the  tail  of  Donati's  comet,  upon  the 


238 


COMETS. 


theory  that  it  was  made  up  of  particles  individually  repelled  by 
the  sun  with  an  intensity  of  force  varying  according  to  this 
law.* 

4O9.  Explanation  of  Situation  and  Curved  Form  of 
Tail.     Let  PC  A  (Fig.  100)  be  a  portion  of  a  comet's  orbit,  the 


FIG.  100. 

sun  being  at  S ;  and  suppose  a  particle  to  be  expelled  in  the 
direction  SAD,  when  the  head  is  at  A,  and  another  particle  to 
be  driven  off  in  the  direction  SEE,  when  the  head  is  at  B.  Each 
particle  will  retain  the  orbital  motion  which  obtained  at  the  time 
of  its  departure,  as  it  moves  away  from  the  sun  ;  and  thus,  when 
the  comet  has  reached  the  point  C,  instead  of  being  at  any  points, 
D  and  E,  on  the  lines  SAD  and  SBE,  will  be  respectively  at 
certain  points,  a  and  6,  farther  forward.  The  line  C6a,  which, 
when  the  comet  is  at  C,  is  the  locus  of  all  the  particles  that  have 
been  emitted  during  the  interval  of  time  in  which  the  comet  has 
been  moving  over  the  arc  AC,  is  the  tail.  We  here  suppose  the 
head  to  be  a  mere  point.  If  we  conceive  the  particles  to  be  con- 
tinually emitted  from  the  marginal  parts  of  the  head,  we  shall 
have  the  hollow  conical  tail  actually  observed.  It  is  easy  to  see 
that  Cfor,  the  line  of  the  tail,  must  be  a  curved  line  concave  to- 
wards the  regions  of  space  which  the  comet  has  left.  Supposing 
the  arc  AC  to  be  so  small,  or  its  curvature  so  slight,  that  it  may 
be  considered  as  a  straight  line,  and  neglecting  the  change  of 
velocity  in  the  orbit,  Ca  will  be  parallel  to  AD,  and  Cb  parallel 
to  BE ;  whence  ECa  =  CSA,  and  KC6  =  CSB.  Thus  the  line 
joining  any  particle  with  the  nucleus  always  makes  an  angle 
with  the  prolongation  of  the  radius-vector,  approximately  equal 

*  See  American  Journal  of  Science,  Yol.  TTTT    pp.  79  and  383,  and  Yol.  xxm 
p.  54,  etc. 


FORMATION   OF  THE  TAILS  OF  COMETS. 

to  the  motion  in  anomaly  during  the  interval  that  has  elapsed 
since  the  particle  left  the  head.  It  follows  from  this,  that  if  we 
suppose  the  velocity  of  the  particles  to  be  continually  the  same, 
and  the  motion  in  anomaly  uniform,  the  deviations  of  the  particles 
a  and  b  from  the  line  of  the  radius- vector  SCR  will  be  in  the 
ratio  of  the  distances  Ca  and  Cb.  But  in  point  of  fact  the  velo- 
city increases  with  the  distance,  so  that  the  curvature  of  the  tail 
will  be  less  than  on  the  supposition  just  made. 

As  to  the  amount  of  the  deviation  of  the  tail  from  the  line  of 
the  radius- vector,  it  must  depend  upon  the  proportion  between 
the  velocity  of  the  particles  and  the  velocity  of  the  head  in  its 
orbit;  and  it  follows  from  the  principle  just  established  that  un- 
less the  velocities  of  emission  augment  as  rapidly  as  the  velocity 
of  revolution,  the  deviation  in  question  will  increase  to  the  peri- 
helion, and  afterwards  decrease,  as  it  is  in  fact  known  to  do. 

41O.  Dispersion  of  the  Comctic  Matter  in  tbc  Plane 
of  the  Orbit.  Observations  made  upon  Donati's  comet,  have 
established  that  the  nebulous  matter  was  much  more  widely  dis- 
persed in  the  direction  of  the  plane  of  the  comet's  orbit  than  in 
the  direction  perpendicular  to  the  orbit;  so  that  the  transverse 
sections  of  the  tail  were  approximately  elliptical  in  form,  and 
more  elongated  in  proportion  as  their  distance  from  the  head 
was  greater.  The  same  fact  was  still  more  conspicuous  in  the 
case  of  the  great  comet  of  1861,  and  is  probably  a  general  law. 
It  is  shown  in  the  memoir  above  referred  to,  that  this  phenome- 
non had  its  origin  in  the  case  of  Donati's  cornet,  in  an  inequality 
in  the  force  of  repulsion  exerted  by  the  sun  upon  different  por- 
tions of  nebulous  matter  expelled  from  the  nucleus.  The  limits 
between  which  the  repulsive  force  varied  were  0  and  1.21  (the 
intensity  of  the  sun's  ordinary  force  of  attraction  at  the  same  dis- 
tance being  the  unit).  It  is  shown  also  that  nearly  one-half  of  the 
tail,  on  the  concave  side,  was  made  up  of  matter  that  was  not 
actually  repelled  by  the  sun,  but  became  widely  separated  from 
the  head  of  the  comet,  after  being  expelled  by  a  projectile  force 
beyond  the  sphere  of  attraction  of  the  nucleus,  simply  because  it 
was  subject  to  a  diminished  intensity  of  solar  attraction.  The 
concave  edge  of  the  tail  consisted  of  matter  subject  to  an  attract- 
ive force  equal  to  •££$  of  the  full  force  of  the  sun's  attraction. 
The  greatest  intensity  of  repulsive  action  (1.21)  obtained  at  the 
convex,  or  preceding  side  of  the  tail. 

If  we  assume  that  the  escaping  particles  did  not  receive  any 
initial  lateral  velocity  from  a  repulsive  or  projectile  force  ex- 
erted by  the  nucleus,  the  limits  of  the  effective  solar  repulsion 
and  attraction,  for  the  two  edges  of  the  tail,  become  1.5  and  0.6 
(instead  of  1.21  and  0.45). 

In  Fig.  101,  the  train  of  the  comet  as  theoretically  determined 
is  compared  with  that  actually  observed.  The  full  curve  runs 
through  the  positions  of  the  particles  that  left  the  head  at  several 


240 


COMETS. 


assumed  dates,  calculated  for  October,  5d.  7h.  mean  time  at 
Greenwich,  and  is  accordingly  the  outline  of  the  train  as  theoreti- 
cally determined  for  that  instant.  The  dotted  curved  line  is  the 


218 


Fia  101. 

outline  of  the  actual  train  as  observed  1£  hours  later,  when  its 
form  and  dimensions  were  sensibly  the  same  as  at  7h.  The  bro- 
ken line  nearly  in  the  middle  of  the  theoretical  train,  runs 
through  the  calculated  positions  of  several  particles  that  left  the 
head  of  the  comet  at  different  dates,  and  were  neither  attracted 
nor  repelled  by  the  sun,  and  therefore  proceeded  on  in  tangents 
to  the  orbit. 

The  single  straight  streamers  seen  in  connection  with  this  and 
other  comets  (See  Plate  III.),  must  have  been  urged  by  a  force 
of  repulsion  many  times  greater  than  the  maximum  limit  of  re- 
pulsion for  the  principal  tail  (1.21). 

II 1.  Columnar  Structure  of  the  Tail  of  Doaiati's  Comet. 
The  tail  of  the  comet  of  Donati.  was  seen  on  certain  occasions  to 
be  traversed,  for  a  part  of  its  length,  by  lands  of  unequal  bright- 
ness, diverging  from  the  vicinity  of  the  head  (See  Plate  III.). 
This  proves  to  have  been  a  consequence  of  frequent  alternations 
in  the  ejection  of  nebulous  matter  from  the  head  of  the  comet; 
for  it  appears,  as  a  result  of  the  calculations  above  mentioned, 
that  all  the  matter  variously  repelled  which  issued  at  any  instant, 
must,  at  any  subsequent  date,  have  been  arranged  nearly  in  a 
straight  line  that  produced  would  pass  near  the  head.  Fig.  102, 
shows,  for  the  date  of  the  calculations  (October  5thX  the  lines 
made  up  of  the  particles  that  proceeded  from  the  nead  of  the 


given,  viz. :  September  29th,  September 
may  accordingly  be  considered  as  having 
series  of  diverging  bands,  or  columns  of 
nebulous  matter  emanating  from  the  head  on  successive  days,  or 
other  equal  intervals  of  time ;  which  alternated  in  brightness 
when  there  were  alternations  in  the  quantity  of  matter  discharged. 


comet,  at  the  dates 
26th,  &c.  The  train 
been  composed  of  a 


CONDITION  AND  ORIGIN   OF  NEBULOUS  ENVELOPES.      241 

412.  The  Source  of  the  Nebulous  Stream,  called  the  tail  of  the 
comet,  has  been  generally  supposed  to  be  the  envelope,  or  enve- 
lopes of  the  head ;  but  at  the  present  day  the  preponderating  weight 


FIG.  102. 

of  evidence  is  opposed  to  this  view,  and  in  favor  of  the  theory  that 
the  envelope  and  tail  are  but  different  portions  of  one  continuous 
stream  of  cometic  matter  emanating  from  the  nucleus,  or  from  the 
bright  nebulosity  contiguous  to  the  nucleus  proper.  It  appears  to 
be  an  insuperable  objection  to  the  former  hypothesis,  that  a  small 
extent  of  the  nebulous  stream,  in  the  immediate  vicinity  of  the 
envelope  from  which  it  proceeds,  contains  as  much  luminous 
matter  as  the  envelope  itself,  and  yet  the  envelope  usually  con- 
tinues in  existence  for  many  days.  Some  of  the  envelopes  that 
were  seen  to  rise  apparently  from  the  nucleus  of  Donati's  Comet, 
did  not  become  dissipated  until  two  weeks  after  their  first  ap- 
pearance. Besides  it  is  certain  that  a  considerable  portion  of  the 
matter  detached  from  the  nucleus,  does  move  in  a  continuous 
stream  through  the  apparent  envelope  into  the  tail ;  for  jets,  or 
single  streams,  are  frequently  seen  to  proceed  from  the  nucleus, 
on  the  side  toward  the  sun,  and  after  being  bent  back  by  the  solar 
repulsion,  to  become  merged  in  the  general  stream  that  seems  to 
issue  from  the  envelope.  It  is  also  possible  to  deduce  the  actual 
form  and  dimensions  of  both  the  envelope  and  tail,  on  the  hy- 
pothesis of  a  single  continuous  stream  proceeding  from  a  certain 
portion  of  the  nucleus  exposed  to  the  action  of  the  sun.* 


CONDITION  AND  ORIGIN  OF  THE  NEBULOUS  ENVELOPES. 

413.  Successive  En rel opes.  As  already  intimated  there  are 
frequently  two  or  more  envelopes  that  appear  to  be  indefinitely 
continued  into  the  train  (See  Plate  III.).  These  are  detached  in 
succession  from  the  nucleus,  and  while  receding  continually 

*  (See  the  American  Journal  of  PCM  nee,  Vol.  XXVII.,  January,  1859.) 

U 


242  COMETS. 

from  it  and  expanding,  decline  in  lustre,  and  finally  disappear; 
according  to  one  of  the  above-mentioned  hypotheses  because  they 
are  dissipated  by  the  repulsive  action  of  the  sun  upon  their  par- 
ticles, and  according  to  the  other  because  the  supply  of  outstream- 
ing  matter  at  the  nucleus  falls  off.  The  late  Director  of  the 
Observatory  of  Harvard  College,  in  his  great  work  on  the  comet 
of  Donati,  states  that  no  less  than  seven  envelopes  were  detached 
in  succession  from  the  nucleus  of  the  comet,  at  intervals  of  from 
four  to  seven  days.  Their  rate  of  recess  from  the  nucleus  was 
about  1,000  miles  per  day.  The  great  comet  of  1861  presented 
a  succession  of  eleven  envelopes,  rising  at  regular  intervals  on 
every  second  day.  Their  evolution  and  final  dissipation  were 
accomplished  with  much  greater  rapidity  than  the  corresponding 
phenomena  of  the  cornet  of  1858. 

414.  Expelling  Force.     Since  the  cometic  particles  which  were  distributed 
along  the  concave  side  of  the  tail  of  Donati's  comet  were  not  repelled  by  the  sun 
(410),  we  must  infer  that  they  were  not  expelled  from  the  nucleus  by  a  force  of 
repulsion,  but  were  in  all  probability  detached  by  some  projectile  force  in  opera- 
tion at  or  near  its  surface.     On  the  other  hand,  the  cometic  particles  that  were  in 
a  condition  to  be  repelled  by  the  sun,  may  have  become  detached  from  the  nucleus 
und<5f*the  operation  of  a  force  of  repulsion  exerted  by  its  mass,  or  from  its  surface. 
We  may  conceive  a  repulsive  force,  exerted,  by  both  the  nucleus  and  the  sun,  to  be 
a  consequence  of  the  particles  being  more  nearly  in  the  condition  of  the  ultimate 
molecules,  in  which  there  is  reason  to  believe  that  they  become  subject  to  both  a 
molecular  and  heat  repulsion,  operating  at  indefinitely  great  distances.* 

If  we  conceive  the  bright  nebulous  mass  adjacent  to  the  nucleus,  which  appears 
to  be  the  fountain  head  of  the  nebulous  stream  that  constitutes  both  the  envelope 
and  train  of  a  comet,  to  be  in  a  magnetic  condition  similar  to  that  which  has  been 
attributed  to  the  photosphere  of  the  sun,  it  is  to  be  observed  that  particles  may 
become  detached  from  the  tops  of  magnetic  columns  simply  in  consequence  of 
a  diminution  in  the  magnetic  intensity  of  the  nucleus  and  its  photosphere ;  and 
such  diminution  of  magnetic  intensity  should  continually  occur,  from  day  to  day, 
as  the  comet  recedes  from  the  sun,  and  consequently  has  a  decreasing  velocity  in 
its  orbit.  For,  according  to  the  theory  of  cosmical  magnetization,  the  intensity  of 
the  magnetic  currents  developed  should  be  directly  proportional  both  to  the  orbital 
velocity  and  the  velocity  of  rotation.!  A  statical  force  of  electric  repulsion  might 
also  operate  to  detach  particles,  whether  magnetic  or  not,  in  directions  normal  to 
the  surface  of  the  nucleus. 

The  Projectile  Force,  whose  existence  we  have  here  recognized,  may  have  its  ori- 
gin in  electric  discharges  along  magnetic  vaporous  columns,  like  the  similar  force 
supposed  to  be  in  action  upon  the  surface  of  the  sun's  photosphere  (293).  In  sup- 
port of  this  view  it  may  be  urged  that,  if  we  assume  the  hypothesis  that  the  nebu- 
lous matter  at  the  nucleus  of  a  comet  is  made  up  of  particles  susceptible  of  mag- 
netization, and  capable  of  being  expelled  by  discharges  along  lines  of  magnetic 
polarization,  we  are  enabled  to  give  an  adequate  explanation  of  diverse  luminous 
phenomena  presented  by  comets,  that  are  wholly  inexplicable  upon  all  previous 
hypotheses. 

415.  Theoretical  Process  of  Evolution  of  an  Envelope.  Wo  would 
first  remark  that  a  rotatory  motion  of  the  nucleus,  in  conjunction  with  its  orbital 
motion,  should,  by  the  collision  of  the  molecules  with  the  ether  of  space,  bring  it 

A  into  a  magnetized  state,  with  the  poles  in  the  vicinity  of  90°  from  the  plane  of  the 
\|  orbit.    Now  if  we  conceive  the  matter,  disposed  in  magnetic  columns,  to  be  ex- 
pelled in  the  lines  of  direction  of  the  columns,  and  subsequently  to  be  repelled  by 
the  sun,  we  have  to  observe  that  the  lines  of  discharge  will  be  nearly  parallel  to 
the  surface  of  the  nucleus  near  the  magnetic  equator,  and  that  their  angle  of  incli- 

*  See  American  Journal  of  Science,  YoL  xxxvm.,  p.  70. 
f  See  American  Journal  of  Science,  Vol.  XLI.,  p.  62. 


ORIGIN  OF  THE  NEBULOUS  ENVELOPES.  243 

nation  to  the  surface  will  increase  with  the  distance  of  the  columns  from  the  mag- 
netic equator,  or  approximately  from  the  plane  of  the  orbit  * 

The  envelope  should  therefore  consist  of  two  portions  proceeding  from  parts  of  the 
nucleus  that  lie  on  opposite  sides  of  the  magnetic  equator.  The  nebulous  streams 
issuing  from  the  points  on  either  side  of  the  equator  will  pass  to  the  other  sicle, 
intersecting  its  plane  at  points  more  and  more  distant  from  the  nucleus,  until  the 
initial  directions  of  the  streams  become,  at  points  at  a  certain  distance  from  the 
equator,  parallel  to  its  plane,  or  to  the  plane  of  the  orbit  nearly.  If  we  conceive 
the  magnetic  equator  to  lie  in  the  plane  of  the  orbit,  and  confine  our  attention  to 
the  streams  proceeding  from  points  on  the  meridian,  whose  plane  contains  the 
radius-vector,  then  at  a  point  on  this  meridian  about  35°  from  the  orbit  the 
nebulous  stream  would  issue  in  a  direction  parallel  to  the  radius-vector,  and  at 
points  that  have  a  higher  magnetic  latitude  its  direction  would  diverge  more  and 
more  from  this  line.  All  such  streams  would  be  bent  back  by  the  force  of  the 
sun's  repulsion,  and  would  form,  collectively,  an  apparent  envelope  on  the  side 
towards  the  sun.  This  would  have  a  parabolic  form,  if  the  discharge  extend  be- 
yond 35°  of  magnetic  latitude,  and  the  expelling  force  and  the  solar  repulsion  have 
each  a  constant  intensity  for  all  latitudes.  But  if  the  latter  should  increase,  or 
the  former  decrease,  with  the  latitude,  the  outline  of  the  envelope  would  approach 
more  nearly  to  the  circular  form,  as  was  observed  in  the  case  of  Donati's  comet. 

416.  Phenomena  confirmatory  of  the  Theory.  Various  peculiarities 
of  form,  and  diversities  of  brightness  presented  by  Donati's  comet,  and  several 
others,  seem  to  indicate  that  each  envelope  does  in  fact  consist  of  two  portions, 
that  do  not  in  general  originate  simultaneously;  and  which  in  part  pass  from 
the  one  side  to  the  other  of  the  nucleus.  The  following  are  some  of  the  peculiar 
features  referred  to : 

(1.)  The  spiral  form,  or  awry  position  of  each  of  the  successive  envelopes,  when 
first  seen  distinctly  separate  from  the  nucleus.  The  explanation  is  that  the  dis- 
charge of  cometic  matter  began  from  the  one  magnetic  hemisphere  sooner  than 
from  the  other ;  from  that  which  is  most  exposed  to  the  sun's  action,  we  may 
suppose. 

(2.)  The  depression,  or  deficiency  of  cometic  matter  about  the  vertex  of  the  enve- 
lope, frequently  noticed,  especially  in  the  later  stages  of  the  envelope.  This  has 
been  in  some  instances  represented  as  a  notch  in  the  envelope.  This  deficiency 
of  light  at  the  vertex  is  obviously  what  should  result  if  the  discharge  should  rela- 
tively fall  off  at  the  magnetic  latitudes  (about  35C)  from  which  the  nebulous  streams 
issue  in  directions  parallel  to  the  radius-vector ;  and  we  shall  soon  see  that  it  is 
reasonable  to  expect  that  the  discharge  should  begin  to  decline  at  these  sooner 
than  at  higher  latitudes. 

(3.)  The  remarkably  dark  land  seen  to  extend  nearly  along  the  axis  of  the  tail 
of  Donati's  comet,  for  a  certain  distance  from  the  nucleus.  This  band  was  too 
dark  to  be  explained  by  the  supposition  that  the  tail  was  hollow.  Upon  the  pre- 
sent theory,  the  brightest  portions  of  the  tail,  near  the  head,  should  have  been  in 
the  plane  perpendicular  to  the  orbit,  and  through  the  radius-vector  :  that  is,  hi  the 
plane  through  the  sun  and  the  magnetic  axis  of  the  nucleus.  A  section  of  the 
tail  and  envelope  in  this  plane  would  show  the  brightest  parts  of  the  two  branches 
streaming  away  from  the  two  magnetic  hemispheres,  on  the  side  towards  the  sun, 
bending  around  past  the  nucleus,  and  separated  there  by  a  dark  space.  In  the 
earlier  and  later  stages  of  an  envelope,  the  dark  shade  would  be  enhanced  by  the 
deficiency  of  the  streams  that  would  return  along  the  axis  (415). 

(4.)  The  great  difference  noticed  in  the  brightness  of  the  two  branches  of  the  train  of 
Donati's  comet,  near  the  head.  This  may  reasonably  be  ascribed  to  an  inequality 
hi  the  discharge  of  nebulous  matter  from  the  two  magnetic  hemispheres.  This 
inequality  of  brightness  was  not  changed  by  the  earth's  passage  through  the  plane 
of  the  comet's  orbit  (on  Sept.  8).  G.  P.  Bond  infers  from  this,  that  "  the  initial 
plane  passing  through  the  two  branches  would  seem  to  have  a  strong  inclination 
to  that  of  the  orbit."  Theory,  as  we  have  seen,  assigns  it  a  position  nearly  per- 
pendicular to  the  plane  of  the  orbit. 

(5.)  The  remarkable  shifting  of  the  superior  brightness  and  eccentric  position  from 
yne  branch  of  the  tail  of  Donati's  comet,  near  the  head,  to  the  other,  about  October  10th. 
At  about  that  date,  the  plane  through  the  sun,  comet,  and  earth,  was  perpendicu- 
lar to  the  plane  of  the  comet's  orbit,  and  the  earth  should  therefore  have  passed 
from  one  side  to  the  other  of  the  initial  plane  of  the  branches  of  the  train.  Cotem- 


244  COMETS. 

poraneously  with  these  changes,  the  dark,  axial  stripe  nearly  disappeared,  and 
reappeared  at  later  dates. 

417.  Explanation  of  the  Rise  and  Recess  of  Successive  Envelopes. 

To  understand  how  one  envelope  after  another  may  rise  and  recede  to  a  cer- 
tain distance  from  the  nucleus,  we  have  to  consider  that  masses  of  vaporous 
magnetic  matter  may  rise,  at  certain  intervals,  from  the  nucleus  to  a  certain  height  in 
its  atmosphere,  under  the  operation  of  the  sun's  rays ;  and  that  such  matter  should 
ascend  most  abundantJy  from  the  equatorial  regions,  where  the  sun  is  supposed  to 
act  most  directly,  and  flow  off  towards  the  poles.  It  will  be  seen,  if  we  consider  the 
diverse  directions  that  would  be  assumed  by  the  magnetic  columns  iu  different  mag- 
netic latitudes,  that,  as  a  necessary  consequence,  the  nebulous  streams  proceeding 
from  them  would  rise-  to  a  greater  and  greater  height  towards  the  sun,  until  the  pro- 
cess of  discharge  reached  the  magnetic  latitude  of  35°.  The  combination  of  all  the 
nebulous  streams  thus  originating  would  present  the  appearance  of  a  luminous 
envelope  on  the  side  of  the  nucleus  towards  the  sun,  the  outer  boundary  of  which 
would  recede  steadily  from  the  nucleus. 

418.  Diversities    in    the    Brightness   of  an    Envelope.     The   great 
diversity  often  observed  in  the  brightness  of  different  parts  of  the  same  envelope, 
may  be  ascribed  to  intersections,  on  the  line  of  sight,  of  the  separate  streams 
of  cometic  particles,   and  to  varying  velocities  in  different  parts  of  the   same 
stream.    Besides  the  ordinary  diversities  which  are  thus  satisfactorily  explained, 
sudden  interruptions  of  brightness  are  often  observed  at  certain  parts  of  an  enve- 
lope ;  these  may  result  from  sudden  variations  in  the  intensity  of  the  expelling 
force,  or  in  the  quantity  of  matter  discharged. 

The  dark  spots  sometimes  seen  are  probably  due  to  a  deficiency  of  nebulous  mat- 
ter near  the  nucleus,  on  certain  magnetic  parallels.  Such  deficiency  may  result 
primarily  from  an  intermission  in  the  ascent  of  nebulous  matter  from  the  equatorial 
regions  of  the  nucleus.  As  the  ascended  matter  flows  off  towards  the  poles,  any 
vacuity  thus  arising  will  gradually  pass  from  one  latitude  to  another,  and  the  spot 
answering  to  it  in  the  envelope  will  rise  and  expand  with  the  envelope. 


THE  FIXED  STABS.  246 


CHAPTER  XIX. 

THE  FIXED  STABS. 
CONSTELLATIONS.— DIVISION  INTO  MAGNITUDES. 

419.  IN  order  to  distinguish  the  fixed  stars  from  each  other, 
they  are  arranged  into  groups,  called  Constellations,  which  are 
imagined  to  form  the  outlines  of  figures  of  men,  animals,  or 
other  objects,  from  which  they  are  named.  Thus,  one  group  is 
conceived  to  form  the  figure  of  a  Bear,  another  of  a  Lion,  a  third 
of  a  Dragon,  and  a  fourth  of  a  Lyre.  The  division  of  the  stars 
into  constellations  is  of  very  remote  antiquity ;  and  the  names 
given  by  the  ancients  to  individual  constellations  are  still  re- 
tained. 

The  resemblance  of  the  figure  of  a  constellation  to  that  of  the 
animal  or  other  object  from  which  it  is  named,  is  in  most  instan- 
ces altogether  fanciful.  Still,  the  prominent  stars  hold  certain 
definite  positions  in  the  figure  conceived  to  be  drawn  on  the  sphere 
of  the  heavens.  Thus,  the  brightest  star  in  the  constellation  Leo 
is  placed  in  the  heart  of  the  Lion,  and  hence  it  has  sometimes 
been  called  Cor  Leonis  or  the  Lion's  Heart:  and  the  brightest 
star  in  the  constellation  Taurus  is  situated  in  the  eye  of  the  Bull, 
and  therefore  sometimes  called  the  BulVs  Eye;  while  that  con- 
spicuous cluster  of  seven  stars  in  this  constellation,  known  by  the 
name  of  the  Pleiades,  is  placed  in  the  neck  of  the  figure.  Again, 
the  line  of  three  bright  stars  noticed  by  every  observer  of  the 
heavens  in  the  beautiful  constellation  of  Orion,  is  in  the  belt  of 
the  imaginary  figure  of  this  bold  hunter  drawn  in  the  skies.  The 
three  Lirger  stars  of  this  constellation  are,  respectively,  in  the 
right  shoulder,  in  the  left  shoulder,  and  in  the  left  foot. 

4*20.  l>if R'reut  Classes  of  Constellations.  The  constella- 
tions are  divided  into  three  classes:  Northern  Constellations, 
Southern  Constellations,  and  Constellations  of  the  Zodiac.  Their  whole 
number  is  91 :  Northern  34,  Southern  45,  and  Zodiacal  12.  The 
number  of  the  ancient  constellations  was  but  48.  The  rest  have 
been  formed  by  modern  astronomers  from  southern  stars  not 
visible  to  the  ancient  observers,  and  others  variously  situated, 
which  escaped  their  notice,  or  were  not  attentively  observed. 

The  zodiacal  constellations  have  the  same  names  as  the  signs  of 
the  zodiac  (Def.  25,  p.  17):  but  it  is  important  to  observe  that  the 
individual  signs  and  constellations  do  not  occupy  the  same  places 


24:6  THE   FIXED  STARS. 

in  the  heavens.  The  signs  of  the  zodiac  coincided  with  the  zo- 
diacal constellations  of  the  same  name,  as  now  defined,  about  the 
year  140  B.  C.  Since  then  the  equinoctial  and  solstitial  points  have 
retrograded  nearly  one  sign :  so  that  now  the  vernal  equinox, 
or  first  point  of  the  sign  Aries,  is  near  the  beginning  of  the  con- 
stellation Pisces;  the  summer  solstice,  or  first  point  of  Cancer, 
near  the  beginning  of  the  constellation  Gemini ;  the  autumnal 
equinox,  or  first  point  of  Libra,  at  the  beginning  of  Virgo;  and 
the  winter  solstice,  or  first  point  of  Capricornus,  at  the  begin- 
ning of  Sagittarius. 

It  follows  from  this,  that  when  the  sun  is  in  the  sign  Aries,  he 
is  in  the  constellation  Pisces,  and  when  in  the  sign  Taurus,  in 
the  constellation  Aries,  &c.  For  the  rest,  it  should  be  ob- 
served that  the  constellations  and  signs  of  the  zodiac  have  not 
precisely  the  same  extent. 

421.  Modes  of  Designation  of   Individual  Stars.     The 
stars  of  a  constellation  are  distinguished  from  each  other  by  the 
letters  of  the  Greek  alphabet,  and  in  addition  to  these,  if  necessary, 
the  Koman  letters,  and  the  numbers  1,  2,  3,  &c. ;  the  characters, 
according  to  their  order,  denoting  the  relative  magnitude  of  the 
stars.     Thus  *  Arietis  designates  the  largest  star  in  the  constella- 
tion Aries ;  /3  Draconis,  the  second  star  of  the  Dragon,  &c. 

Some  of  the  fixed  stars  have  particular  names,  as  Siriiis,  Alde- 
baran,  Arcturus,  Regulus,  &c. 

422.  Magnitudes.     The  stars  are  also  divided  into  classes, 
or  magnitudes,  according  to  the  degrees  of  their  apparent  bright- 
ness.    The  largest  or  brightest  are  said  to  be  of  the  first  magni- 
tude; the  next  in  order  of  brightness,  of  the  second  magnitude; 
and  so  on   to  stars  of  the  sixth  magnitude,  which  includes  all 
those  that  are  barely  perceptible  to  the  naked  eye.     All  of  a 
smaller  kind  are  called  telescopic  stars,  being  invisible  without  the 
assistance  of  the  telescope.     The  classification  according  to  ap- 
parent magnitude  is  continued  with  the  telescopic  stars  down  to 
stars  of  the  twentieth  magnitude  (according  to  Sir  John  Herschel), 
and  the  twelfth  according  to  Struve. 

The  following  are  all  the  stars  of  the  first  magnitude  that  oc- 
cur in  the  heavens,  viz. :  Sirius  or  the  Dog-star,  Betelgeux,  Rigel, 
Aldebaran,  Capella,  Procyon,  Regulus,  Denebola,  Cor.  Hydrce, 
iSpica  Virginis,  Arcturus,  Antares,  Altair,  Vega,  Deneb  or  Alpha 
Oygni,  Dubhe  or  Alpha  Ursce  Majoris,  Alpherat  or  Alpha  Andro- 
medai,  Fomalhaut,  Achernar,  Canopus,  Alpha  Crucis,  and  Alpha 
Centauri.  It  is  the  practice  of  Astronomers  to  mark  more  or 
less  of  these  stars  as  intermediate  between  the  first  and  the  sec- 
ond magnitude ;  and  in  some  catalogues  some  of  them  are  as- 
signed to  the  second  magnitude.  All  of  these  stars,  with  the 
exception  of  the  last  four,  come  above  the  horizon  in  all  parts  of 
the  United  States. 

423.  Celestial  Globe.     There  are  two  principal  modes  of 


CONSTELLATIONS. — DIVISION  INTO   MAGNITUDES.          247 

representing  the  relative  positions  of  the  stars;  the  one  by 
delineating  them  on  a  globe,  where  each  star  occupies  the  spot 
in  which  it  would  appear  to  an  eye  placed  in  the  centre  of  the 
globe,  and  where  the  situations  are  reversed  when  we  look  down 
upon  them ;  the  other  is  by  a  chart  or  map,  where  the  stars  are 
generally  so  arranged  as  to  be  represented  in  positions  similar  to 
their  natural  ones,  or  as  they  would  appear  on  the  internal  con- 
cave surface  of  the  globe.  The  construction  of  a  globe  or  chart, 
is  effected  by  means  of  the  right  ascensions  and  declinations  of 
the  stars.  Two  points  diametrically  opposite  to  each  other  on 
the  surface  of  an  artificial  globe  are  taken  to  represent  the  poles 
of  the  heavens,  and  a  circle  traced  90°  distant  from  these  for  the 
equator :  another  point  23£°  from  one  of  the  poles  is  then  fixed 
upon  for  one  of  the  poles  of  the  ecliptic,  and  with  this  point  as 
a  geometrical  pole  a  great  circle  described  ;  the  points  of  inter- 
section of  the  two  circles  will  represent  the  equinoctial  points. 
The  point  which  represents  the  place  of  a  star  is  found  by  mark- 
ing off  the  right  ascension  and  declination  of  the  star  upon  the 
globe. 

All  the  fixed  stars  visible  to  the  naked  eye,  together  with  some 
of  the  telescopic  stars,  are  represented  on  celestial  globes  of  12 
or  18  inches  in  diameter. 

494.  Catalogue  of  Stan.  The  places  of  the  fixed  stars  are 
generally  expressed  by  their  right  ascensions  and  declinations, 
but  sometimes  also  by  their  longitudes  and  latitudes.  A  table 
containing  a  list  of  fixed  stars  designated  by  their  proper  char- 
acters, and  giving  their  mean  right  ascensions  and  declinations, 
or  their  mean  longitudes  and  latitudes,  is  called  a  Catalogue  of 
those  stars.  (See  Table  XC  (a)  ). 


NUMBER  AND  DISTRIBUTION  OVER  THE  HEAVENS. 

425.  The  number  of  stars  visible  to  the  naked  eye,  in  the  en- 
tire sphere  of  the  heavens,  is  from  6,000  to  7,000 ;  of  which 
nearly  4,000  are  in  the  northern  hemisphere;  but  not  more  than 
2,000  can  be  seen  with  the  naked  eye  at  any  one  hour  of  the 
night  at  a  given  place.  The  telescope  brings  into  view  many 
millions,  and  every  material  augmentation  of  its  space-penetrat- 
ing power  greatly  increases  the  number. 

As  to  the  number  of  stars  belonging  to  each  different 
magnitude,  astronomers  assign  from  20  to  24  to  the  first  magni- 
tude, from  50  to  60  to  the  second,  about  200  to  the  third,  and  so 
on  ;  the  numbers  increasing  very  rapidly  as  we  descend  in  the 
scale  of  brightness;  the  whole  number  of  stars  already  registered 
down  to  the  seventh  magnitude,  inclusive,  amounting  to  12,000 
or  15,000. 

The  reason  of  this  increase  in  the  number  of  the  stars,  as  we 


248  THE   FIXED  STARS. 

descend  from  one  magnitude  to  another,  is  undoubtedly  that  in 
general  the  stars  are  less  bright  in  proportion  as  their  distance  is 
greater ;  while  the  average  distance  between  contiguous  stars  is 
about  the  same  for  one  magnitude  as  for  another.  It  is  easy  to 
see  that  upon  these  suppositions  the  number  of  stars  posited  at 
any  given  distance,  and  having  therefore  the  same  apparent  mag- 
nitude, will  be  greater  in  proportion  as  this  distance  is  greater, 
and  thus  as  the  apparent  magnitude  is  lower. 

426.  Comparative  Brightness.     It  is  not  to  be  understood 
that  the  classification  of  the  stars  into  different  magnitudes,  is 
made  according  to  any  fixed  definite  proportion  subsisting  be- 
tween the  degrees  of  apparent  brightness  of  the  stars  belonging 
to  different  classes.     Stars  of  almost  every  gradation  of  bright- 
ness, between  the  highest  and  the  lowest,  are  met  with.     Those 
which  offer  marked  differences  of  lustre,  form  the  basis  of  the 
classification ;  others,  which  do  not  differ  very  widely  from  these, 
are  united  to  them.    As  a  necessary  consequence,  there  are  some 
stars  of  intermediate  lustre,  which  cannot  be  assigned  with  cer- 
tainty to  either  magnitude.     Thus,  in  the  catalogue  published 
by  the  Astronomical  Society  of  London,  3  stars  are  marked  as 
intermediate  between  the  first  and  second  magnitudes,  and  29 
between  the  second  and  third. 

Different  astronomers  also  not  ^infrequently  assign  the  same 
star  to  different  magnitudes. 

As  to  the  proportions  of  light  emitted  from  the  average  stars 
of  the  different  magnitudes,  according  to  the  experimental  com- 
parisons of  Sir  Wm.  Herschel,  they  are,  from  the  first  to  the 
sixth  magnitude,  approximately  in  the  ratio  of  the  numbers, 
100,  25,  12,  6,  2,  1. 

427.  Distribution  of  the  Stars.     With   the  exception   of 
the  three  or  four  brightest  classes,  the  stars  are  not  distributed  in- 
discriminately over  the  sphere  of  the  heavens,  but  are  accumu- 
lated in  far  greater  numbers  on  the  borders  of  that  belt  of  cloudy 
light  in  the  heavens,  which  is  called  the  milky  way,  and  in  the 
milky  way  itself,  which  the  telescope  shows  to  consist  of  an  im- 
mense number  of  stars  of' small  magnitude  in  close  proximity. 
According  to  Struve,  the  total  number  of  stars  visible  in  the 
Herschelian  telescope  of  20  feet  focus  and  19  inches  aperture,  is 
a  little  over  2'»,000,000. 

42§.  Stratum  of  the  Milky  Way.  The  great  accumula- 
tion of  stars  in  a  zone  of  the  heavens,  encompassing  the  earth  in 
the  direction  of  a  great  circle,  suggested  to  the  mind  of  Herschel 
the  idea  that  the  stars  of  our  firmament  are  not  disseminated 
indifferently  throughout  the  surrounding  regions  of  space,  but 
are  for  the  most  part  arranged  in  a  stratum,  the  thickness  of 
which  is  very  small  in  comparison  with  its  breadth;  the  sun 
and  solar  system  being  near  the  middle  of  the  thickness.  If  S 
(Fig.  103)  represents  the  place  of  the  sun,  it  will  be  seen  that 


NUMBER  AND   DISTRIBUTION  OVER  THE  HEAVENS.       249 


FIG.  103. 

side  of  the  point  S,  the  stratum  is 
certain  distance  into  two  laminae, 
as  shown  in  the  figure,  which  re- 
presents a  section  of  the  supposed 
stratum.  This  supposition  is  ne- 
cessary to  account  for  the  two 
branches,  with  a  dark  space  be- 
tween them,  into  which  the  milky 
way  is  divided  for  about  one-third 
of  its  course. 

Herschel  undertook  to  gauge  this  stra- 
tum in  various  directions,  on  the  principle 
that  the  distance  through  to  its  borders  in 
any  direction  was  greater  in  proportion  as 
the  number  of  stars  seen  in  that  direction 
was  greater.  He  thus  found  that  its  actu- 
al form  was  very  irregular ;  its  section,  in- 
stead of  being  truly  that  of  a  segment  of  a 
sphere  divided  for  a  certain  distance  into 
two  lamrinse,  as  represented  in  Fig.  103, 
having  the  form  represented  in  Fig.  104. 
He  estimated  the  thickness  of  the  stratum 
to  be  less  than  160  times  the  interval  be- 
tween the  stars,  and  the  breadth  to  be  no- 
where greater  than  1,000  times  the  same 
distance.  These  are  his  first  results ;  we 
shall  see  in  the  sequel  that  they  were 
materially  modified  by  his  subsequent  in- 
vestigations. 

Sir  John  Herschel  conceives  that 
the  superior  brilliancy  and  larger 
development  of  the  milky  way 
in  the  southern  hemisphere,  from 
the  constellation  Orion  to  that  of 
Antinous,  indicate  that  the  sun 
and  his  system  are  at  a  distance 
from  the  centre  of  the  stratum  in 
the  direction  of  the  Southern 
Cross,  and  that  the  central  parts 
are  so  vacant  of  stars  that  the 
whole  approximates  to  the  form 
of  an  annulus. 


upon  this  supposition 
the  number  of  stars  in 
the  direction  SO  of  the 
thicknessof  the  stratum, 
will  be  less  than  in  any 
other  direction,  and  that 
the  greatest  n  umber  will 
lie  in  the  direction  of  the 
breadth,  as  SB.  On  one 
supposed  to  be  divided  for  a 


Fid.  104. 


250 


THE  FIXED  STARS. 


ANNUAL  PARALLAX  AND  DISTANCE  OF  THE  STARS. 

429.  The  Annual  Parallax  of  a  fixed  star  is  the  angle  made 
by  two  lines  conceived  to  be  drawn,  the  one  from  the  sun  and 
the  other  from  the  earth,  and  meeting  at  the  star,  at  the  time  the 
earth  is  in  such  part  of  its  orbit  that  its  radius- vector  is  perpen- 
dicular to  the  latter  line ;  or,  in  other  words,  it  is  the  greatest 
angle  that  can  be  subtended  at  the  star  by  the  radius  of  the  earth's 
orbit.  Thus,  let  S  (Fig.  105)  be  the  sun,  s  a  fixed  star,  and  E 

the  earth,  in  such  a  position 
that  the  radius -vector  SE  is 
perpendicular  to  Es  the  line 
of  direction  of -the  star,  then 
the  angle  SsE  is  the  annual 
parallax  of  the  star  s. 

43O.  Least  Distance  of 
the  Stars.  If  the  annual  par- 
allax of  a  star  were  known, 
we  might  easily  find  its  dis- 
tance from  the  earth ;  for  in 
the  right-angled  triangle  SEs 

FIG.  105.  we  would  know  the  angle 

SsE  and  the  side  SE,  and 

we  should  only  have  to  compute  the  side  Es.  Now,  if  any  of 
the  fixed  stars  have  a  sensible  parallax,  it  could  be  detected  by 
a  comparison  of  the  places  of  the  star,  as  observed  from  two 
positions  of  the  earth  in  its  orbit,  diametrically  opposite  to  each 
other;  and  accordingly,  the  attention  of  astronomers  furnished 
with  the  most  perfect  instruments,  has  long  been  directed  to  such 
observations  upon  the  places  of  some  of  the  fixed  stars,  in  order 
to  determine  their  annual  parallax.  But,  after  exhausting  every 
refinement  of  observation,  they  have  not  been  able  to  establish, 
until  quite  recently,  that  any  of  them  have  a  measurable  paral- 
lax. Now,  such  is  the  nicety  to  which  the  observations  have 
been  carried,  that,  did  the  angle  in  question  amount  to  as  much 
as  1",  it  could  not  possibly  have  escaped  detection  by  the  methods 
of  observation  employed.  We  may  then  conclude  that  the  an- 
nual parallax  of  the  nearest  fixed  star  is  less  than  V. 

Taking  the  parallax  at  1",  the  distance  of  the  star  comes  out 
206,265  times  the  distance  of  the  sun  from  the  earth,  or  about 
20  millions  of  millions  of  miles.  The  distance  of  the  nearest 
fixed  star  must  therefore  be  greater  than  this.  A  juster  notion 
of  the  immense  distance  of  the  fixed  stars,  than  can  be  conveyed 
by  figures,  may  be  gained  from  the  consideration  that  light, 
which  traverses  the  distance  between  the  sun  and  earth  in 
8m.  18s.,  and  would  perform  the  circuit  of  our  globe  in  £  of  a 
second,  employs  3J  years  in  coming  from  the  nearest  fixed  star 
to  the  earth. 


ANNUAL  PARALLAX  AND  DISTANCE  OF  THE  STARS.      251 


431.  Determination  of  the  Parallax  of  a  Fixed  Star. 

The  long  continued  endeavor  to  detect  an  annual  parallax  of  a 
fixed  star,  by  the  direct  method  of  comparing  the  places  of  the 
star,  determined  at  an  interval  of  half  a  year,  has  at  last  been 
crowned  with  success.  The  parallax  of  a  Centauri  has  been  thus 
determined  by  Professor  Henderson,  from  observations  made  in 
1832  and  1833,  with  a  large  mural  circle.  Subsequent  observa- 
tions with  a  more  efficient  instrument  by  Maclear  have  furnished 
an  angle  of  parallax  differing  but  little  from  that  obtained  by 
Henderson.  Its  value  is  0".913,  which  answers  to  a  distance 
about  ^j-  less  than  the  least  limit  of  distance  of  the  stars,  just 
determined.  The  parallax  and  distance  of  Sirius  and  of  the  pole- 
star,  have  since  been  determined  in  a  similar  manner,  but  with 
less  certainty.  The  result  obtained  for  the  parallax  of  the 
pole-star  is  (/Ml,  and  for  that  of  Sirius  an  angle  a  little  greater. 
A  parallax  of  0".ll  answers  to  a  distance  that  light  would  re- 
quire nearly  30  years  to  traverse. 

432.  Parallax  of  a  Star  found  by  the  Differential  method.     The 

honor  of  being  the  first  to  determine  with  certainty  the  parallax  and  distance  of 
a  fixed  star  belongs  to  BesseL  The  star  observed  by  him  is  that  designated  as  61 
CygnL  It  is  a  star  of  about  the  6th  magnitude,  barely  visible  to  the  naked  eye. 
When  viewed  through  a  telescope  it  is  seen  to  consist  of  two  stars  of  nearly 
equal  brightness,  at  a  distance  from  each  other  of  about  16".  These  stars  have  a 
motion  of  revolution  around  each  other,  and  the  two  move  together  at  the  same 
rate  of  5".3  per  year,  as  one  star,  along  the  sphere  of  the  heavens.  It  is  hence 
inferred  that  they  are  bound  together  into  one  system  by  the  principle  of  gravita- 
tion, and  are  at  pretty  nearly  the  same  distance  from  the  eartbl  The  great  proper 
motion  of  this  double  star,  as  compared  with  other  stars,  led  to  the  suspicion  that 
it  was  nearer  than  any  other ;  and  thus  to  attempts  to  determine  its  parallax.  The 
principle  of  Bessel's  method  is  to  find  the  difference  between  the  parallaxes  of  the 
star  61  Cygni,  and  some  other  star  of  much  smaller  magnitude,  and  therefore  sup- 
posed to  be  at  a  much  greater  distance,  seen  in  as  nearly  the  same  direction  as  pos- 
sible. This  difference  will  differ  from  the  absolute  parallax  of  the  double  star  by 
only  a  small  fraction  of  its  whole  amount.  It  was  found  by  measuring  with  a 
position  micrometer  (62)  the  annual  changes  in  the  distance  of  the  two  stars,  and 
in  the  position  of  the  line  joining  them.  To  make  it  evident  that  such  changes 
will  be  an  inevitable  consequence  of  any  difference  of  parallax  in  the  two  stars, 
conceive  two  cones  haviug  the  earth's  orbit 
for  a  common  base,  and  their  vertices  respec- 
tively at  the  two  stars,  and  imagine  their  sur- 
faces to  be  produced  past  the  stare  until  they 
intersect  the  heavens.  The  intersections  will 
be  ellipses,  but,  by  reason  of  the  different 
distances  of  the  two  stars,  of  different  sizes, 
as  represented  in  Fig.  106 ;  and  they  will  be 
apparently  described  by  the  stars  in  the 
course  of  one  revolution  of  the  earth  in  its 
orbit  The  two  stars  will  always  be  similarly 
situated  in  their  parallactic  ellipses :  thus, 
if  one  is  at  A  the  other  will  be  at  a ;  and 
after  the  earth  has  made  one-quarter  of  a 
revolution,  they  will  be  at  B  and  6;  and  after 
another  quarter  of  a  revolution  at  C  and  c,  Ac. 
Now  it  will  be  manifest,  on  inspecting  the 
figure,  the  ellipses  being  of  unequal  size,  PiO.  106. 

that  the  lines  of  the   stars  will  be  of  un- 
equal lengths,  and  have  different  directions  in  the  different  situations  of  the  stars. 


252  THE   FIXED  STARS. 

A  much  smaller  angle  of  parallax  maybe  found,  with  the  same  degree  of  certainty, 
by  this  indirect  method,  than  by  the  direct  process  explained  in  Art.  430;  for  since 
the  two  stars  are  seen  in  pretty  nearly  the  same  direction,  they  will  be  equally 
affected  by  refraction  and  aberration ;  and  since  it  is  only  the  relative  situations 
of  the  two  stars  that  are  measured,  no  allowance  has  to  be  made  for  precession 
and  nutation,  or  for  errors  in  the  construction  or  adjustment  of  the  instrument 
It  is  therefore  independent  of  the  errors  that  are  inevitably  committed  in  the 
determination  of  these  several  corrections,  when  it  is  attempted  to  find  directly 
the  absolute  parallax,  by  observing  the  right  ascension  and  declination  at  oppo- 
site seasons  of  the  year.  The  measurements  made  with  the  micrometer  hi  the 
hands  of  the  most  accurate  observers,  may  be  relied  on  as  exact  to  within  a  small 
fraction  of  1". 

For  the  sake  of  greater  certainty  Bessel  made  the  measurements  of  parallactic 
changes  of  relative  situation  between  the  star  61  Cygni  and  two  small  stars  in- 
stead of  one, — the  middle  point  between  the  two  members  of  the  double  star  being 
taken  for  the  situation  of  this  star.  He  found  the  difference  of  parallax  to  be  for 
the  one  star  0".3584,  and  for  the  other  star  0".3289 :  and  assuming  the  absolute 
parallax  of  the  two  stars  to  be  equal,  found  for  the  most  probable  value  of  the  dif- 
ference of  parallax  0  ".3483.  Whence  he  calculated  the  distance  of  the  star  61 
Cygni  to  be  592,200  times  the  mean  distance  of  the  earth  from  the  sun;  a  distance 
which  would  be  traversed  by  light  in  9^  years. 

The  number  of  stars  whose  parallax  and  distance  have  been 
determined,  more  or  less  accurately,  by  both  methods,  now 
amounts  to  12.  The  least  parallax  obtained  is  that  of  Capella, 
which  is  0".05  »  but  it  must  be  regarded  as  quite  uncertain. 

433.  Comparative*  Distances  of  Stars  of  Different 
Magnitudes.  According  to  Peters,  the  mean  parallax  of  stars 
of  the  second  magnitude  is  0".116,  which  answers  to  a  distance 
that  light  would  traverse  in  28  years.  From  this  result  the 
mean  parallax  and  distance  of  stars  of  each  of  the  different 
magnitudes  have  been  approximately  deduced  by  means  of  the 
relative  distances  of  stars  of  the  different  magnitudes,  as  de- 
termined by  Struve  on  the  assumption  that  the  stars  are  uni- 
formly distributed  through  space  (at  least  in  certain  directions), 
and  that  the  light  from  the  stars  of  the  different  magnitudes 
varies  according  to  a  certain  admitted  law.  The  mean  distance  of 
stars  of  the  first  magnitude,  as  computed,  is  traversed  by  light  in 
15*5  years ;  and  that  of  a  star  of  the  sixth  magnitude  in  120  years. 
Light  requires  138  years  to  come  from  the  most  remote  star  visible 
to  the  naked  eye.  The  same  principle  of  computation  of  distances 
being  extended  to  the  telescopic  stars,  it  appears  that  the  stars 
just  visible  in  the  Herschelian  telescope  of  20  ft.  focus  are  sep- 
arated from  us  by  a  distance  that  light  takes  3, 500  years  to  jour- 
ney over.  This  is  on  the  supposition  that  the  rays  of  light  do 
not  experience  any  sensible  degree  of  extinction  in  traversing 
the  regions  of  space. 


NATURE  AND  MAGNITUDE  OF  THE  STARS. 

' 

434.  The  vast  distance  at  which  the  fixed  stars  are  visible,  and 
shine  with  a  light  not  much  inferior  to  the  planets,  leaves  no 
room  to  doubt  that  they  are  all  suns,  or  self-luminous  bodies. 


VARIABLE  STARS.  253 

if  it  should  be  conjectured  that  some  of  the  fainter  stars  might 
be  bodies  shining  by  reflected  light,  like  the  planets,  the  answer 
is,  that  if  we  were  to  suppose  the  existence  of  opake  bodies,  at 
the  distance  of  the  stars,  so  inconceivably  vast  in  their  dimen- 
sions as  to  send  a  sensible  light  to  the  eye,  if  illuminated  to  the 
same  degree  as  the  planets,  the  stars  of  the  smaller  magnitudes 
are  too  remote  from  the  brighter  ones  to  receive  sufficient  light 
from  them ;  for,  the  smallest  measurable  space  in  the  field  of  the 
larger  telescopes  is,  at  the  distance  of  the  nearer  stars,  nearly 
as  large  as  the  earth's  orbit.  It  is  perhaps  possible,  that  some 
of  the  faintest  members  of  some  of  the  double  stars,  as  surmised 
by  Sir  John  Herschel,  may  shine  by  reflected  light. 

435.  Magnitude  of  the  Stars.  To  be  able  to  determine  the 
magnitude  of  a  star,  we  must  know  its  distance,  and  also  its  ap- 
parent diameter.  Now  the  distances  of  but  few  stars  have  as 
yet  been  found ;  and  the  discs  of  all  the  stars,  even  in  the  most 
powerful  telescopes,  are  altogether  spurious ;  so  that  in  no  in- 
stance have  we  the  data,  nor  have  we  reason  to  expect  that  they 
will  be  hereafter  obtained,  for  determining  with  certainty  the 
magnitude  of  a  fixed  star. 

But  we  may  infer  from  the  quantity  of  their  light  as  compared 
with  that  of  the  sun,  and  the  mean  distances  of  stars  of  the  dif- 
ferent magnitudes,  as  approximately  determined  (433),  that  the 
stars  are  in  general  much  larger  than  the  sun.  According  to 
the  mean  result  of  recent  photometrical  comparisons  made  by 
Messrs.  G.  P.  Bond  and  Alvan  Clark,  between  the  bright  star  * 
Lyrse  and  the  sun,  if  the  sun  were  removed  to  133,500  times  its 
present  distance  it  would  send  us  the  same  quantity  of  light  as 
this  star.  From  this  we  may'infer  that  if  it  were  removed  to 
the  distance  of  the  nearest  star  (430),  it  would  appear  as  a  star 
of  the  second  magnitude ;  and  that  if  it  were  removed  to  the 
mean  distance  of  stars  of  the  first  magnitude,  it  would  appear 
as  a  star  of  the  sixth  magnitude,  and  be  just  visible  to  the  naked 
eye.  It  would  seem  then  that  the  sun  is  much  smaller  than  most, 
if  not  all,  of  the  stars  of  the  first  magnitude ;  and  is  presumably 
also  smaller  than  most  of  the  stars  of  the  other  magnitudes. 


VARIABLE  STARS. 

436.  There  are  many  stars  which  exhibit  periodical  changes 
of  brightness ;  these  are  termed  Variable  Stars.  More  than  a 
hundred  stars  are  now  known  to  belong  to  this  class.  One  of 
the  most  remarkable  of  the  variable  stars  is  o  Ceti,  or  Mira. 
From  being  as  bright  as  a  star  of  the  second  magnitude,  it  gradu- 
ally decreases  until  it  entirely  disappears ;  and  after  remaining 
for  a  time  invisible,  reappears,  and  gradually  increasing  in  lustre, 
finally  recovers  its  original  appearance.  The  mean  period  of 


254:  THE   FIXED   STARS. 

these  changes  is  33 1£  days.  The  star  remains  at  its  greatest 
brightness  about  two  weeks,  employs  about  three  months  in 
waning  to  its  disappearance,  continues  invisible  for  about  five 
months,  and  during  the  remaining  three  months  of  its  period 
increases  to  its  original  lustre.  Such  has  been  the  general  course 
of  its  phases.  But  it  does  not  always  recover  the  same  degree 
of  brightness,  nor  increase  and  diminish  by  the  same  grada- 
tions. It  is  even  related  by  Hevelius,  that  in  one  instance  it  re- 
mained invisible  for  a  period  of  four  years.  A  similar  phenome- 
non has  been  noticed  in  the  case  of  the  star  ^  Cygni.  According 
to  the  testimony  of  Cassini,  it  was  scarcely  visible  throughout 
the  years  1699,  1700,  and  1701,  at  those  times  when  it  should 
have  been  most  conspicuous.  On  the  other  hand  a  variable  star 
situated  in  the  Northern  Crown,  sometimes  fluctuates  in  its 
brightness  very  slightly  for  several  years,  and  then  suddenly  re- 
sumes its  regular  variations,  in  the  course  of  which  it  entirely 
disappears. 

The  greater  number  of  variable  stars  undergo  a  regular  in- 
crease and  diminution  of  lustre  without  ever  becoming  entirely 
invisible.  Algol,  or  0  Persei,  is  a  remarkable  variable  star  of 
this  description.  For  a  period  of  2d.  14h.,  it  appears  as  a  star  of 
the  second  magnitude,  after  which  it  suddenly  begins  to  dimin- 
ish in  splendor,  and  in  about  3J  hours  is  reduced  to  a  star  of 
the  fourth  magnitude.  It  then  begins  again  to  increase,  and  in 
3£  hours  more  is  restored  to  its  usual  brightness,  going  through 
all  its  changes  in  2d.  20h.  49m. 

Besides  the  single  variable  stars,  there  are  also  a  number  of 
double  stars,  one  or  both  the  members  of  which  are  variable ; 
as  7  Virginis,  f  Arietis,  £  Bootis,  &c. 

437.  General  Facts.  Two  general  facts  have  been  noticed 
with  respect  to  the  variable  stars  which  are  worthy  of  remark, 
viz. :  that  the  color  of  their  light  is  red,  and  that  their  period  of 
increase  of  lustre  is  shorter  than  that  of  the  decrease.  The  star 
Algol,  offers  an  exception  to  both  of  these  general  facts.  The 
ruddy  color  is  especially  observable  in  the  case  of  the  smaller 
variable  stars.  It  is  a  curious  and  suggestive  fact  that  a  number 
of  the  variable  stars  present  a  hazy  appearance  at  their  min- 
imum, as  if  some  form  of  nebulous  matter  were  interposed  be- 
tween them  and  the  eye. 

43§.  Temporary  Stars.  There  are  also  some  instances  on 
record  of  temporary  stars  having  made  their  appearance  in  the 
heavens ;  breaking  forth  suddenly  in  great  splendor,  and  without 
changing  their  positions  among  the  other  stars,  after  a  time  en- 
tirely disappearing.  One  of  the  most  noted  of  these  is  the  star 
which  suddenly  shone  forth  with  great  brilliancy  on  the  llth  of 
November,  1572,  between  the  constellations  Cepheus  and  Cassio- 
peia, and  was  attentively  observed  by  Tycho  Brahe*.  It  was 
then  as  bright  as  any  of  the  permanent  stars,  and  continued  to 


VARIABLE   STARS.  255 

increase  in  splendor  till  it  surpassed  Jupiter  when  brightest,  and 
was  visible  at  mid-day.  It  began  to  diminish  in  December  of 
the  same  year,  and  in  March,  1574,  entirely  disappeared,  after 
having  remained  visible  for  sixteen  months,  and  has  not  since 
been  seen. 

It  was  noticed  that  while  visible  the  color  of  its  light  changed  from  white  to 
yellow,  and  then  to  a  very  distinct  red ;  after  which  it  became  pale,  like  Saturn. 

In  the  years  945  and  1264,  brilliant  stars  appeared  in  the  same 
region  of  the  heavens.  It  is  conjectured  from  the  tolerably  near 
agreement  of  the  intervals  of  the  appearance  of  these  stars,  and 
that  of  1572,  that  the  three  may  be  one  and  the  same  star,  with  a 
period  of  about  300  years.  The  places  of  the  stars  of  945  and 
1264  are,  however,  too  imperfectly  known  to  establish  this  with 
any  degree  of  certainty. 

Besides  these  three  temporary  stars,  several  others  have  made  their  appearance. 
viz. :  one  in  the  year  134  B.  C.,  seen  by  Hipparchus ;  another  in  389  A.  D.,  in  the 
constellation  Aquila;  a  third  in  the  9th  century,  in  Scorpio;  a  fourth  in  1604,  in 
Serpentarius,  seen  by  Kepler  ;  a  fifth  in  1670,  in  the  Swan ;  and  a  sixth  in  1848, 
in  Ophiuchus. 

What  is  no  less  remarkable  than  the  changes  we  have  noticed, 
several  stars,  which  are  mentioned  by  the  ancient  astronomers, 
have  now  ceased  to  be  visible,  and  some  are  now  visible  to  the 
naked  eye  which  are  not  in  the  ancient  catalogues.  • 

439.  Explanation  of  Variable  Stars.  The  most  probable  explanation  of 
the  phenomenon  of  variable  stars  is  that  they  are  self-luminous  bodies  rotating 
upon  axes,  and  having,  like  the  sun,  spots  developed  periodically  on  their  surface, 
under  the  varying  action  of  revolving  planets  upon  their  photospheres.  The  range 
of  the  planetary  action  must  be  regarded  as  much  greater  than  in  the  case  of  the 
sun.  •  The  fluctuations  generally  observable  in  the  periods  and  in  the  maxima  and 
minima  of  brightness  of  the  variable  stars,  are  analogous  to  the  fluctuations  that 
occur  in  the  periods  and  maxima  and  minima  of  the  sun's  spots.  Prof.  Wolf  has 
minutely  investigated  this  correspondence  of  phenomena,  in  the  case  of  certain 
stars,  by  constructing  curves  showing  their  variations  of  light  in  detail.  The 
hazy  appearance  often  presented  by  variable  stars  at  their  minimum,  may  result 
from  the  interposition  of  nebulous  matter  expelled  from  the  star  in  the  process  of 
formation  of  the  spots  on  its  surface  (293).  The  ruddy  color  frequently  noticed 
may  be  ascribed  to  a  lower  temperature  consequent  upon  a  greater  prevalence 
of  spots,  or  to  more  intense  electric  discharges  within  the  photosphere. 

In  the  case  of  the  star  Algol  the  phenomena  are  precisely  such  as  would  result 
from  the  periodical  interposition  of  an  opake  body.  In  those  cases  in  which  the 
period  of  the  diminution  of  the  light  is  a  large  fraction  of  the  entire  period  of  the 
star,  as  well  as  those  in  which  there  are  occasional  interruptions  in  the  regular 
recurrence  of  the  phenomena,  the  supposition  of  the  interposition  of  an  opake 
body  will  not  answer. 

Temporary  stars  may  be  supposed  to  be  suns  which  have  entirely  omitted  the 
evolution  of  light  for  a  long  period  of  time,  and  then  burst  forth  anew  with  a  sud- 
den and  peculiar  splendor,  under  the  influence  of  a  planetary  action  retaining  to 
its  maximum  at  the  end  of  a  long  period.  Or  they  may  possibly  result  from  an 
encounter  of  two  stars  at  the  point  of  intersection  of  the  vast  orbits  which  they 
pursue  in  the  regions  of  space.  The  remarkable  fact,  noticed  by  Sir  John  Her- 
schel,  that  all  the  temporary  stars  on  record,  of  which  the  places  are  distinctly 
indicated,  have  occurred  in  or  close  upon  the  borders  of  the  Milky  Way,  where, 
as  we  shall  see,  the  stars  are  most  condensed,  lends  some  support  to  the  latter 
hypothesis. 


256  THE  FIXED  STARS. 


DOUBLE  STARS. 

44O.  Many  of  the  stars  which  to  the  naked  eye  appear  single, 
when  examined  with  telescopes  are  found  to  consist  of  two  (in 
some  instances  three  or  more)  stars  in  close  proximity  to  each 
other.  These  are  called  Double  Stars,  or  Multiple  Stars.  (See  Fig. 
107.)  This  class  of  bodies  was  first  attentively  observed  by  Sir 
William  Herschel,  who,  in  the  years  1782  and  1785,  published 


Castor.      y  Leonis.       Rigel.        Pole-star.     11  Monoc.    £  Cancri. 

FIG.  lot. 

catalogues  of  a  large  number  of  them  which  he  had  observed. 
The  list  has  since  been  greatly  increased  by  Professor  Struve,  of 
Dorpat,  Sir  J.  F.  W.  Herschel,  and  other  observers,  and  now 
amounts  to  several  thousand. 

441.  J>egree  of  Proximity.     Double  stars  are  of  various 
degrees  of  proximity.      In  a  great  number  of  instances,  the 
angular  distance  of  the  individual  stars  is  less  than  1",  and  the 
two  can  only  be  separated   by  very  powerful  telescopes.      In 
other  instances,  the  distance  is  •£'  and  more,  and  the  separation 
can  be  effected  with  telescopes  of  very  moderate  power.     They 
are  divided  into  different  classes  or  orders,  according  to  their 
distances ;  those  in  which  the  proximity  is  the  closest  forming 
the  first  class. 

442.  Comparative  Size.     The  two  members  of  a  double 
star  are  generally  of  quite  unequal  size  (See  Fig.  107).     But  in 
some  instances,  as  that  of  the  star  Castor,  they  are  of  nearly  the 
same  apparent  magnitude.     Double  stars  occur  of  every  variety 
of  magnitude.     Sirius  is  the  largest  of  the  double  stars.     It  is 
attended  by  a  minute  companion  star,   at  a  distance  of  10". 
This  was  first  discovered  by  Clark,  with  his  great  telescope  of 
18£in.  aperture. 

In  some  instances  one  of  the  constituents  of  a  double  star  is 
itself  double,  e  Lyrse  offers  the  remarkable  combination  of  a 
double-double  star. 

443.  Different  Colors.    It  is  a  curious  fact,  that  the  two  constituents  of  a  dou- 
ble star  in  numerous  instances  shine  with  different  colors ;    and  it  is   still  more 
curious  th,at  these  colors  are  in  general  complementary  to  each  other.    Thus,  the 
larger  star  is  usually  of  a  ruddy  or  orange  hue,  while  the  smaller  one  appears 
blue  or  green.    This  phenomenon  has  been  supposed  to  be  in  some  cases  the 
effect  of  contrast ;  the  larger  star  inducing  the  accidental  color  in  the  feebler  light  of 
the  other.    Sir  John  Herschel  cites  as  probable  examples  of  this  effect  the  two 
stars  i  Cancri,  and  y  Andromedae.    But  it  is  maintained  by  Nichol  that  this  expla- 
nation cannot  be  admitted ;    for,  if  true,  it  ought  to  be  universal,  whereas  there 
are  many  systems  similar  in  relative  magnitudes  to  the  contrasted  ones,  in  which 


DOUBLE  STARS.  257 

both  stars  are  yellow,  or  otherwise  belong  to  the  red  end  of  the  spectrum. 
Again,  if  the  blue  or  violet  color  were  the  effect  of  contrast,  it  ought  to  disap- 
pear when  the  yellow  star  is  hid  from  the  eye ;  which,  however,  it  does  not  do. 
Thus,  the  star  &  Cygni  consists  pf  two  stars,  of  which  one  is  yellow,  and  the  other 
shines  with  an  intensely  blue  light ;  and  when  one  of  them  is  concealed  from 
view  by  an  interposed  slip  of  darkened  copper,  the  other  preserves  its  color 
unchanged.  The  color,  then,  of  neither  of  the  stars  can  be  accidental. 

It  may  be  remarked  in  this  connection,  that  the  isolated  stars  also  shine  with 
various  colors.  For  example,  among  stars  of  the  first  magnitude,  Sirius,  Vega, 
Altair,  Spica  are  white,  Aldebaran,  Arcturus,  Betelgeux  red,  Capella  and  Proey- 
on  yellow.  In  smaller  stars  the  same  difference  is  seen,  and  with  equal  distinct- 
ness when  they  are  viewed  through  telescopes.  According  to  Herschel,  insulated 
stars  of  a  deep  red  color,  occur  in  many  parts  of  the  heavens,  but  no  decidedly 
green  or  blue  star  has  ever  been  noticed  unassociated  with  a  companion  brighter 
than  itself. 

444.  Discovery  of  Binary  Stars.  Sir  William  Herschel 
instituted  a  series  of  observations  upon  several  of  the  double 
stars,  with  the  view  of  ascertaining  whether  the  apparent  relative 
situation  of  the  individual  stars  experienced  any  change  in  con- 
sequence of  the  annual  variation  of  the  parallax  of  the  star. 
With  a  micrometer  adapted  to  the  purpose,  (62),  he  measured 
from  time  to  time  the  apparent  distance  of  the  two  stars,  and 
the  angle  formed  by  their  line  of  junction  with  the  meridian  at 
the  time  of  the  meridian  passage,  called  the  Angle  of  Position. 
Instead,  however,  of  finding  that  annual  variation  of  these  angles, 
which  the  parallax  of  the  earth's  annual  motion  would  produce, 
he  observed  that,  in  many  instances,  they  were  subject  to  regu- 
lar progressive  changes  which  seemed  to  indicate  a  real  motion 
of  the  stars  with  respect  to  each  other.  After  continuing  his 
observations  for  a  period  of  twenty-five  years,  he  satisfactorily 
ascertained  that  the  changes  in  question  were  in  reality  produced 
by  a  motion  of  revolution  of  one  star  around  the  other,  or  of 
both  around  their  common  centre  of  gravity  ;  and  in  two  papers, 
published  in  the  Philosophical  Transactions  for  the  years  1803 
and  1804,  he  announced  the  important  discovery  that  there  ex- 
ist sidereal  systems  composed  of  two  stars  revolving  about  each 
other  in  regular  orbits.  These  stars  have  received  the  appel- 
lation of  Binary  Stars,  to  distinguish  them  from  other  double 
stars  which  are  not  thus  physically  connected,  and  whose  appar- 
ent proximity  may  be  Occasioned  by  the  circumstance  of  their 
being  situated  on  nearly  the  same  line  of  direction  from  the  earth, 
though  at  very  different  distances  from  it.  Similar  stars,  con- 
sisting of  more  than  two  constituents,  are  called  Ternary,  Quater- 
nary, &c. 

Since  the  time  of  Sir  W.  Herschel,  the  observations  upon  the 
binary  stars  have  been  'continued  by  Sir  John  Herschel,  Sir 
James  South,  Struve,  Bessel,  Madler,  and  other  astronomers. 
According  to  Madler  the  number  of  known  binary  and  ternary 
stars  is  now  about  600.  Every  year  materially 'increases  the 
list ;  and  will  probably  continue  to  do  so  for  some  time  to  come : 
for,  while  the  changes  of  relative  situation  are  in  some  instances 

17 


258  THE   FIXED  STARS. 

exceedingly  slow,  the  actual  number  of  such  systems  is  probably 
a  large  fraction  of  the  whole  number  of  double  stars  ;  at  least,  if 
we  confine  our  attention  to  double  stars  whose  constituents  are 
within  %'  of  each  other.  This  may  be  inferred  from  the  fact, 
that  the  number  of  such  double  and  multiple  stars  actually  ob- 
served, which  amounts  to  over  3000,  is  at  least  ten  times  greater 
than  the  number  of  instances  of  fortuitous  juxtaposition  that 
would  obtain  on  the  supposition  of  a  uniform  distribution  of  the 
stars.  Besides,  there  are  a  number  of  double  stars  not  yet  dis- 
covered to  have  a  motion  of  revolution,  which  still  give  indica- 
tions of  a  physical  connection.  Thus,  their  constituents  are 
found  to  have  constantly  the  same  proper  motion  in  the  same 
direction ;  showing  that  they  are  in  all  probability  moving  as 
one  system  through  space. 

445.  Periods  and  Orbits  of  Binary  Stars.  From  the  ob- 
servations made  upon  some  of  the  binary  stars,  astronomers  have 
been  enabled  to  deduce  the  form  of  their  orbits,  and  approxi- 
mately the  lengths  of  their  periods.  The  orbits  are  ellipses  of 
considerable  eccentricity.  The  periods  are  of  various  lengths,  as 
will  be  seen  from  the  following  enumeration  of  some  of  those 
considered  as  best  ascertained:  M-2  Bootis  650  years,  y  Virginia 
171  years,  p  Ophiuchi  92  years,  *  Centauri  77  years,  ^Cancri  58 
years,  £  Herculis  36  years.  Fig.  108  represents  a  portion  of  the 


Fig.  108. 

apparent  orbit  of  the  double  star  y  Yirginis,  and  shows  the  rela- 
tive positions  of  the  two  members  of  the  double  star  in  various 
years.  At  the  time  of  their  nearest  approach,  in  1836,  the  inter- 
val between  them  was  a  fraction  of  1",  and  they  could  not  be 
separated  by  the  best  telescopes,  with  a  magnifying  power  of 
1000.  Since  then  their  distance  has  been  continually  increasing. 
In  1844  it  amounted  to  2",  and  a  power  of  from  200  to  300 
was  sufficient  to  separate  them.  The  orbit  represented  in  the 


PROPER  MOTIONS  OF  THE  STARS.  259 

figure  is  the  stereographic  projection  of  the  true  orbit  on  a  plane 
perpendicular  to  the  line  of  sight. 

The  actual  distance  between  the  members  of  a  binary  star  has 
been  found  for  61  Cygni,  and  *  Centauri.  Bessel  makes  it  for 
the  first  about  two  and  a  half  times  the  distance  of  Uranus  from 
the  sun. 

It  is  important  to  observe,  that  the  revolution  of  one  star 
around  another  is  a  different  phenomenon  from  the  revolution 
of  a  planet  around  the  sun.  It  is  the  revolution  of  one  sun 
around  another  sun ;  of  one  solar  system  around  another  solar 
system  ;  or  rather  of  both  around  tneir  common  centre  of  gravi- 
ty. We  learn  from  it  the  important  fact,  that  the  fixed  stars  are 
endued  with  the  same  property  of  attraction  that  belongs  to  the 
sun  and  planets. 

PROPER  MOTIONS  OF  THE  STARS. 

446.  It  has  already  been  stated  that  the  fixed  stars,  so  called, 
are  not  all  of  them  rigorously  stationary.     By  a  careful  compari- 
son of  their  places,  found  at  different  times  with  the  accurate  in- 
struments and  refined  processes  of  modern  observation,  it  has 
been  found  that  great  numbers  of  them  have  a  progressive  mo- 
tion along  the  sphere  of  the  heavens,  from  year  to  year.     The 
velocity  and  direction  of  this  motion  are  uniformly  the  same  for 
the  same  star,  but  different  for  different  stars.     One  of  the  stars 
which  has  the  greatest  proper  motion,  is  the  double  star  61  Cyg- 
ni.    During  the  last  fifty  years  it  has  shifted  its  position  in  the 
heavens  4' 21";  the  annual  proper  motion  of  each  of  the  indi- 
vidual stars  being  5".2.     An  isolated  star,  called  *  Indi,  has  a 
still  greater  proper  motion.    It  changes  its  place  7". 7  every  year. 
The  proper  motions  of  some  of  the  stars  are  either  partially  or 
entirely  attributable  to  a  motion  of  the  sun  and  the  whole  solar 
sj'stem  in  space;  but  the  motions  of  others  cannot  be  reconciled 
with  this  hypothesis,  and  must  be  regarded  as  indicative  of  a  real 
motion  of  these  bodies  in  space. 

447.  rttolioii  of  the  Solar  Sy§teni  through  Space.    The 
first  successful  attempt  to  explain  the  proper  motions  of  the  fixed 
stars  on  the  hypothesis  of  a  motion  of  the  solar  system  through 
space,  was  made  by  Sir  William  Herschel.     After  a  careful  ex- 
amination of  these  motions,  he  conceived  that  the  majority  of 
them  could  be  explained  on  the  supposition  of  a  general  recess 
of  the  stars  from  a  point  near  that  occupied  by  the  star  A  Hereu- 
lis  towards  a  point  diametrically  opposite.     Whence  he  inferred 
that  the  sun,  with  its  attendant  system  of  planets,  was  moving 
rapidly  through  space  in  a  direction  towards  this  constellation. 
Doubt  has  since  been  thrown  upon  these  conclusions  by  Bessel 
and  other  astronomers ;  but  they  have  recently  been  decisively 
reestablished  by  M.  Argelander,  of  Abo.     The  investigations  of 


260  THE   FIXED  STABS. 

Argelander,  which  were  communicated  to  the  Academy  of  St. 
Petersburgh  in  1837,  have  since  been  confirmed  by  M.  Otto 
Struve,  of  the  Pulkowa  Observatory,  and  other  eminent  ob- 
servers. 

Taking  the  mean  of  all  the  more  recent  determinations,  we 
find  the  most  probable  situation  of  the  point  towards  which  the 
sun's  motion  is  directed  to  be  as  follows :  E.  A.  260°  14',  K  Dec. 
35°  10'.  This  point  is  a  little  to  the  east  and  north  of  the  star  u 
in  the  constellation  Hercules,  and  about  9°  distant  from  the  point 
first  supposed  by  Herschel. 

44§.  Velocity  of  Smi'§  Motion  through  Space.  0.  Struve 
finds  that  for  a  star  situated  at  right  angles  to  the  direction  of 
the  sun's  motion,  and  placed  at  the  mean  distance  of  the  stars  of 
the  first  magnitude,  the  annual  angular  displacement  due  to  the 
sun's  motion  is  0".339  (with  a  probable  error  of  0".02o).  So 
that,  if  we  assume,  according  to  the  best  determinations,  0".209 
for  the  hypothetical  value  of  the  parallax  of  a  star  of  the  first 
magnitude,  it  follows  that  at  the  distance  of  the  star  supposed 
the  annual  motion  of  the  sun  subtends  an  angle  1.623  times 
greater  than  the  radius  of  the  earth's  orbit :  which  makes  it 
nearly  150,000,000  of  miles.  This  is  at  the  rate  of  4.7  miles 
per  second. 

449.  Velocity  of  the  proper  motions  of  the  stars.     The  above 
angle  of  0".339  is  about  the  greatest  annual  displacement  which 
a  star   can    experience   in  consequence   of  the  sun's   motion. 
Whence  it  appears  that  the  whole  of  the  proper  motion  of  any 
star  which  is  over  and  above  this  amount  must  certainly  be  due 
to  a  real  motion  in  space.     Thus,  in  the  case  of. the  star  61 
Cygni,  nearly  5"  of  its  annual  proper  motion  (5".23)  must  result 
from  an  actual  motion  in  space.     This  is  14.37  times  greater 
than  the  parallax  of  this  star  (0".35).     Accordingly,  if  we  sup- 
pose the  direction  of  its  motion  to  be  perpendicular  to  its  line 
of  direction  from  the  sun  or  earth,  its  annual  motion  is  14.37 
times  greater  than  the  radius  of  the  earth's  orbit,  or  at  the  rate 
of  nearly  42  miles  per  second.     As  we  have  no  means  of  ascer- 
taining the  actual  direction  of  its  motion,  it  is  impossible  to  dis- 
cover how  much  the  velocity  exceeds  this  determination. 

450.  Sun's   motion  comparatively  slow.      By  comparing  the 
particular  motions  presented  by  stars  of  different  classes  _ with 
the  motion  of  the  solar  system,  viewed  perpendicularly  at  the 
distance  of  a  star  of  the  first  magnitude,  as  above  given,  it  is 
found  that  the  former,  at  the  mean,  are  2.4  times  greater  than 
that  of  the  sun ;  whence  it  follows  that  this  luminary  may  be 
ranked  among  those  stars  which  have  a  comparatively  slow 
motion  in  space. 


NEBUUE.  261 


CLUSTERS  OF  STARS. 

451.  In  many  parts  of  the  heavens  stars  are  seen  crowded 
together  into  clusters,  often  in  numbers  too  great  to  be  counted. 
Some  of  these  clusters  are  visible  to  the  naked  eye.     One  of  the 
most  conspicuous  is  that  called  the  Pkiades.     To  the  unaided 
sight  it  appears  to  consist  of  six  or  seven  stars,  but  with  a  tele- 
scope of  moderate  power  50  or  60  conspicuous  stars  are  seen 
grouped  together  within  the  same  space,  and  more  than  100 
smaller  ones  are  distinctly  discernible. 

In  the  constellation  Cancer  is  a  luminous  spot  called  Prcesepe, 
or  the  bee-hive,  which  a  telescope  of  moderate  power  resolves 
entirely  into  stars.  Within  a  space  about  £°  square,  more  than 
40  conspicuous  stars  are  seen,  besides  many  smaller  ones.  In  the 
sword-handle  of  Perseus  is  another  cloudy  spot  thickly  Crowded 
with  stars,  which  become  separately  visible  with  a  telescope  of 
low  power. 

One  of  the  richest  clusters  in  the  northern  hemisphere  occurs 
in  the  constellation  Hercules,  between  the  stars  tj  and  s.  It  is 
visible  to  the  naked  eye,  on  clear  nights,  as  a  hazy  mass  of  light ; 
which  is  readily  resolved  into  stars  by  a  good  telescope.  Viewed 
through  a  telescope  of  high  power  it  presents  the  magnificent 
aspect  of  an  innumerable  host  of  stars  crowded  together  towards 
the  centre  into  a  perfect  blaze  of  light. 

The  richest  and  largest  cluster  in  the  whole  heavens  is  seen  in 
the  constellation  Centaurus,  in  the  southern  hemisphere.  It  is 
visible  to  the  naked  eye  as  a  nebulous  star,  and  is  designated  u 
Centauri.  The  telescope  shows  it  to  consist  of  an  immense  mul- 
titude of  stars  congregated  together  in  the  form  of  a  magnificent 
globular  cluster  (see  Fig.  1,  Plate  IV.).  In  the  field  of  view  of 
a  large  telescope,  it  has  an  apparent  diameter  nearly  equal  to 
that  of  the  moon. 

NEBULAE. 

452.  With  the  aid  of  the  telescope,  a  great  number  of  faintly 
luminous  spots,  or  patches,  are  seen  scattered  here  and  there  over 
the  sphere  of  the  heavens.     These  are  called  Nebulae.     Some  of 
these  nebulous  objects  are  partially  visible  to  the  naked  eye,  but 
the  great  majority  of  them  cannot  be  discerned  without  the  assis- 
tance of  a  good  telescope,  and  very  many  are  beyond  the  reach 
of  any  but  the  most  powerful  instruments. 

453.  Number  and  Distribution  of  Nebulae.     The  num- 
ber of  nebulae  hitherto  discovered,  is  over  5,000.     They  are  very 
unequally  distributed  over  the  heavens,  especially  in  the  north- 
ern hemisphere.     They  are  most  abundant  in  the  constellations 
Virgo,  Leo,  Coma  Berenices,  Canes  Venatici,  and  Ursa  Major ; 


262  THE   FIXED  STARS. 

and  occur  in  astonishing  profusion  in  certain  regions  in  this 
quarter  of  the  heavens,  as  in  the  northern  wing  of  Virgo.  When 
the  telescope  is  directed  towards  these  regions  it  is  observed  that 
the  nebulae  follow  each  other  in  rapid  succession,  from  the  diur- 
nal motion  of  the  heavens  ;  while,  in  some  parts  of  the  heavens, 
hours  elapse  after  one  of  them  has  passed  through  the  field  before 
another  enters.  In  the  southern  hemisphere  there  are  two  de- 
tached spaces  of  considerable  extent,  visible  to  the  naked  eye, 
called  the  Magellanic  Clouds,  that  shine  with  a  nebulous  light 
like  the  milky  way,  which  are  thickly  sown  with  nebulae. 

454.  Diversity  of    Form    and  Appearance.      As  seen 
through  telescopes  of  moderate  power,  the  nebulse  are,  for  the 
most  part,  round  or  oval  in  form ;  but,  when  carefully  examined 
with  the  larger  telescopes,  they  are  found  to  present  a  great 
variety  of  aspects  and  forms.     A  large  number  are  found  to 
consist  of  a  multitude  of  minute  stars  distinctly  separate,  and 
condensed  about  one  or  more  points  within  the  mass.     Many 
others  take  on  the  appearance  of  incipient  resolvability,  charac- 
terized by  the  phrase  star-dust,  and  are  doubtless  real  clusters 
too  distant,  or  too  much  condensed,  to  show  their  individual 
stars.     Others  still  offer  no  appearance  of  stars,  and  remain  the 
same  cloud-like  objects  when  the  highest  telescopic  power  is  ap- 
plied to  them.     These  Irresolvable  Nebulce  were  supposed  by  Sir 
William  Herschel  to  be  masses  of  actual  nebulous  matter  dis- 
seminated through  space,  but  are  now  generally  believed  to  be 
clusters,  or  beds  of  stars,  like  the  rest ;  only  too  vastly  remote 
to  be  revealed  as  such  by  any  optical  means  yet  employed. 

455.  Classification  of  Nebulae.     The  nebulse  are  classified 
according  to  their  aspects  and  forms,  as  seen  through  the  larger 
telescopes,  as  follows :  (1)  Globular  Clusters,  (2)  Irregular  Clus- 
ters, (3)   Oval  Nebulce,  (4)  Annular  Nebulce,  (5)  Planetary  Nebulce, 
(6)  Stellar  Nebulce,  (7)  Spiral  Nebulce,  (8)  Irregular  Nebulce,  (9) 
Double  Nebulce. 

456.  Ulohular  Clusters  take  their  name  from  their  sup- 
posed actual  form.      Their  component   stars   are  so  crowded 
together  as  to  form  an  almost  definite  outline,  and  they  run  up  to 
a  blaze  of  light  towards  the  centre,  where  their  condensation  is  the 
greatest.     The  number  of  stars  congregated  in  a  single  cluster 
is  to  be  told  only  by  thousands  and  tens  of  thousands ;  although 
their  apparent  size  does  not  in  any  instance  exceed  the  -fa  part 
of  the  moon's  disc.     They  are,  in  general,  difficult  of  resolution, 
and  appear  in  telescopes  of  moderate  power  as  small,  round,  nebu- 
lous specks,  resembling  a  comet  without  a  tail.     Fig.  3,  Plate 
IV.,  represents  a  globular  cluster  to  be  seen  in  the  constellation 
Pegasus. 

457.  Irregular  Clusters.     These  are  more  or  less  irregular 
and  indefinite  in  their  outline.     They  are  generally  less  rich  in 
stars,  and  less  condensed  towards  the  centre  than  the  globular 


NEBULA.  263 

clusters.  Fig  2,  Plate  IV.,  represents  an  irregular  cluster  situated 
in  the  constellation  Capricornus.  The  Pleiades,  and  Coma  Bere- 
nices, are  instances  of  irregular  clusters  whose  individual  stars 
are  seen  in  telescopes  of  low  power. 

Irregular  clusters  occur  of  almost  every  degree  of  condensation, 
from  a  cluster  which  seems  to  be  only  a  space  of  an  irregular 
and  ill-defined  outline,  somewhat  more  rich  in  stars  than  the  sur- 
rounding regions,  to  the  perfectly  defined  globular  cluster  highly- 
condensed  at  the  centre. 

458.  Oval  Nebulae.     Nebulae  having  a  distinct  elliptic  out- 
line occur  of  various  degrees  of  eccentricity,  from  moderately 
oval  to  an  elongation  almost  linear  (see  Figs.  5,  6,  and  7,  Plate 
IV.).    They  are  more  condensed,  though  in  very  different  degrees, 
in  their  central  parts,  and  often  present  great  and  sudden  varia- 
tions of  brightness  from  one  portion  of  their  mass  to  another. 

This  is  very  observable  in  Fig.  9,  Plate  V.  Such  nebulae,  which 
retain  their  oVal  form  in  the  field  of  the  most  powerful  telescope, 
are  doubtless  spheroidal. clusters,  in  their  general  form,  though 
more  or  less  complex  in  their  internal  structure.  Many  of  them 
are  either  wholly  or  partially  resolvable  into  individual  stars. 
Others  afford  to  the  eye  only  indistinct  intimations  of  their  stel- 
lar structure.  In  general  the  spheroidal  clusters  are  far  more 
difficult  of  resolution  than  globular  clusters. 

459.  Dynamical  Equilibrium  of  Sidereal  Systems.     It 
cannot  be  doubted  that  the  systematic  organization  of  sidereal 
systems  has  been  determined  under  the  operation  of  the  princi- 
ple of  universal  gravitation ;    and  it  is  plain  that  in   the  in- 
stance of  globular   and  spheroidal  clusters,  a  general  state  of 
equilibrium  would  be  possible  only  upon  the  supposition  that  the 
individual  stars  of  each  cluster  revolve  around  some  central 
point.     Such  a  general  dynamical  equilibrium  of  a  cluster  may 
however  exist,  and  the  internal  structural  condition  be  subject 
at  the  same  time  to  secular  changes,  from  the  varying  combina- 
tions of  individual  orbital  positions,  and  the  disturbing  actions 
of  some  of  the  component  stars  on  one  another. 

460.  Annular  Nebulae.    A  very  small  number  of  observed 
nebulae  have  the  annular  form  (Fig.  12,  Plate  Y.).     A  conspicu- 
ous example  of  this  singular  class  of  nebulae  may  be  seen  with 
a  telescope  of  moderate  power,  midway  between  the  stars  £  and 
v  Lyrae.     The  central  vacuity  is  not  perfectly  dark,  but  filled 
with  a  faint  nebulous  light.     The  telescope  of  Lord  Eosse,  has 
resolved  it  into  minute  stars,  and  shown  it  to  be  fringed  on  its 
outer  edge  with  filaments  of  stars  (Fig.  11,  Plate  Y.).   Chacornac, 
of  the  Paris  observatory,  describes  it  as  presenting,  in  Foucault's 
great  telescope  of  plated  glass,  the  appearance  of  a  hollow  cylin- 
drical bed  of  very  small  stars,  with  a  thin  stratum  of  minute 
stars  stretching  across  the  centre. 

461.  Planetary  Nebulae   have  a  round  planet-like  disc  of 


264  THE  FIXED  STARS. 

an  equable  light  throughout,  or  only  slightly  mottled,  and  often 
perfectly  definite  in  outline.  As  many  as  25  of  these  curious 
objects  have  been  discovered.  A  large  planetary  nebula  occurs 
near  /8  Ursae  Majoris.  It  is  nearly  3'  in  diameter.  There  is  a 
still  larger  planetary  nebula  in  the  constellation  Bootes.  If  we 
suppose  the  former  nebula  to  be  at  no  greater  distance  than  * 
Ceritauri,  the  nearest  fixed  star,  its  linear  diameter  must  still  be 
more  than  three  times  the  diameter  of  the  orbit  of  Neptune. 
Its  actual  distance  must  be  vastly  greater  than  here  supposed,  and 
its  dimensions  correspondingly  greater,  unless  its  individual  stars 
are  very  minute  in  comparison  with  the  most  distant  isolated  stars. 

If  we  suppose  them  to  be  of  the  same  size  as  the  more  distant 
stars,  its  distance  should  be  equally  great,  and  its  dimensions 
more  than  1,000  times  greater  than  the  above  determination. 

One  of  the  planetary  nebulae  has  been  resolved  by  Lord 
Eosse's  telescope,  and  another  shown  to  be  an  annular  nebula. 
This  class  of  nebulae  are  generally  supposed  to  be  either  cylindri- 
cal beds  of  stars,  or  assemblages  of  stars  in  the  form  of  hollow 
spherical  shells. 

462.  Stellar  Nebulae  are  those  in  which  one  or  more  stars 
are  seen  apparently  connected  with  a  nebulosity.  This  class  of 
nebulae  comprises  several  varieties,  the  most  important  of  which 
is  that  of  the  Nebulous  Stars.  Nebulous  stars  are  stars  encircled 
by  a  faint  nebulosity ;  in  some  cases  terminating  in  a  distinct 
outline,  in  others  shading  off  gradually  into  the  general  light  of 
the  sky  (Fig.  14,  Plate  V.).  Fig.  16,  rlate  V.,  shows  the  appear- 
ance of  a  nebulous  star  in  Gemini,  as  seen  through  Lord  Kosse's 
telescope.  The  stars  surrounded  by  these  nebulous  atmo- 
spheres have  the  same  appearance  as  other  stars;  and  their 
atmospheres  offer  no  indication  of  resolvability  into  stars  with 
the  most  powerful  telescopes. 

Fig.  15,  Plate  V.,  is  a  remarkable  stellar  nebula  in  the  con- 
stellation Cygnus.  It  consists  of  a  star  of  the  llth  magnitude, 
surrounded  by  a  very  bright  and  perfectly  round  planetary 
nebula  of  uniform  light,  nearly  15'  in  diameter.  Herschel 
regards  it  as  constituting  a  connecting  link  between  planetary 
nebulse  and  nebulous  stars. 

In  the  other  varieties  of  stellar  nebulse  stars  are  seen  occupy- 
ing various  positions,  in  apparent  connection  with  nebulous 
masses  which  are  generally  of  an  oval  form.  Sometimes  the 
nebulosity  is  spindle-shaped,  with  a  star  at  each  end.  One 
variety  has  received  the  name  of  Cometic  Nebulce,  from  their 
close  resemblance  to  a  comet  with  a  spreading  tail.  Fig.  18, 
Plate  V.,  represents  a  cometic  nebula  in  the  tail  of  Scorpio. 

463.  Spiral  Nebulae.  The  great  telescope  of  Lord  Kosse 
has  revealed  the  remarkable  fact  that  some  of  the  nebulae  are 
made  up  of  spiral  convolutions  proceeding  from  a  common 
jmeleus,  or  from  two  nuclei.  The  most  conspicuous  example  of 


NEBULA.  265 

this  curious  form  is  represented  in  Fig.  10,  Plate  Y.  It  is 
situated  near  the  star  ij,  at  the  extremity  of  the  tail  of  the  Great 
Bear.  The  spiral  nebulous  coils  diverge  from  two  bright  cen- 
tres, about  5'  apart.  As  seen  in  the  field  of  his  great  reflecting 
telescope,  they  are  described  by  Lord  Rosse  as  "  breaking  up 
into  stars."  Another  beautiful  spiral  nebula  is  situated  in  the 
northern  wing  of  Yirgo.  In  some  of  the  instances  cited  by 
Lord  Rosse,  the  spiral  arrangement  was  only  partially  made  out. 

464.  Irregular  Nebulae.  Under  this  head  are  classed  all 
the  remaining  single  nebulae  that,  as  seen  through  the  best  tele- 
scopes, have  no  simple  geometrical  form.  The  majority  of  these 
are  of  great  extent  in  comparison  with  other  nebulae,  and  are 
devoid  of  all  symmetry  of  form.  They  are  also  remarkable  for 
the  great  irregularities  observable  in  the  distribution  of  their 
light,  indicating  a  singular  complexity  of  internal  structure. 

Ttie  Great  Nebula  in  the  sword  handle  of  Orion  is  the  most  con- 
spicuous example  of  this  class  of  nebulae.  It  consists  of  irregu- 
lar nebulous  patches  extending  over  a  surface  about  40'  square, 
or  about  twice  the  size  of  the  moon's  disc.  From  its  great  mag- 
nitude and  beauty,  singularly  grotesque  form,  and  the  wonderful 
variety  of  its  light,  it  is  the  most  remarkable  of  all  the  nebulae. 
One  portion,  near  the  trapezium  or  sextuple  star  d,  is  uncom- 
monly bright,  and  is  visible  to  the  naked  eye.  Other  portions 
are  quite  hazy  and  dim ;  and  still  other  intervening  parts  are 
dark,  and  even  absolutely  black.  Sir  John  Herschel  describes 
the  brightest  portions  as  resembling  the  head  and  yawning  jaws 
of  some  monstrous  animal,  with  a  sort  of  proboscis  running  out 
from  the  snout.  The  constitution  of  this  singular  nebula 
remained  enveloped  in  mystery  from  the  time  of  its  first  dis- 
covery by  Huyghens,  in  1 656,  until  the  telescope  of  Lord  Rosse 
was  directed  upon  it ;  when  the  brighter  portion  near  the  tra- 
pezium was  distinctly  perceived  to  consist  of  clustering  stars. 
The  elder  Bond,  with  the  great  Cambridge  refractor,  subsequently 
succeeded  in  resolving  the  same  part  of  the  nebula.  More 
recently  G.  P.  Bond  has  detected  indications  of  an  arrangement 
of  the  separated  stars  in  spiral  lines. 

The  Great  Nebula  in  Andromeda  is  another  remarkable  irregu- 
lar nebula.  In  the  field  of  the  Cambridge  telescope  it  has  the 
irregular  outline  and  peculiar  appearance  represented  in  Fig.  8, 
Plate  IV.  Its  extreme  length  is  2£°,  and  breadth  over  1°.  It 
is  traversed,  for  a  considerable  portion  of  its  length,  by  "two 
dark  bands  or  canals."  Certain  parts  offered,  in  the  same  tele- 
scope, decided  indications  of  a  stellar  constitution.  The  brighter 
portion  of  this  nebula  is  distinctly  visible  to  the  naked  eye. 
As  viewed  with  a  telescope  of  moderate  power,  it  has  an  elon- 
gated oval  form,  similar  to  Fig.  7,  Plate  IV. 

The  Crab  Nebula.  Fig.  4,  Plate  IY.,  represents  the  appear- 
ance of  this  curious  nebula  as  seen  through  Lord  Rosse's  tele- 


266  THE   FIXED  STARS. 

scope.  It  is  described  as  studded  with  stars,  mixed  with  a 
nebulosity  probably  consisting  of  stars  too  minute  to  be  recog- 
nised, and  exhibiting  filaments  extending  out  from  the  southern 
portion  of  the  nebula.  In  ordinary  telescopes  these  outlying 
branches,  which  have  suggested  the  name  of  crab  nebula,  are 
invisible,  and  the  part  seen  has  an  oval  form. 

The  Dumb-bell  Nebula  is  so  named  from  the  fact  that  as  seen 
through  a  telescope  of  moderate  size,  in  which  the  brighter 
portion  alone  is  visible,  it  has  the  apparent  form  of  a  dumb-bell. 
In  Lord  Rosse's  telescope  the  nebula  appears  less  regular  in  its 
form;  and  it  is  at  the  same  time  seen  to  consist  of  innumerable 
stars  mixed  with  irresolvable  nebulosity.  When  its  fainter  por- 
tions are  included,  its  outer  limit  has  an  oval  form  (see  Fig.  9, 
Plate  V.),  which  shows  the  nebula  as  viewed  through  the  smaller 
telescope  of  3  feet  aperture,  constructed  by  Lord  Rosse. 

465.  Double  Nebulae.  A  considerable  number  of  double 
nebulae  occur  in  different  parts  of  the  heavens.  M.  D?  Arrest, 
of  Copenhagen,  enumerates  fifty  whose  constituents  are  not  over 
5'  apart,  and  estimates  that  there  may  be  as  many  as  200  such 
double  nebulae.  The  two  constituents  are  most  commonly  circu- 
lar in  their  apparent  form,  and  are  probably  real  globular  clusters. 
(Fig.  17,  Plate  V.) 

The  individual  members  of  most  of  these  nebulae  are  probably 
physically  connected.  In  one  instance  considerable  changes 
have  been  recognised  in  the  distance  arid  relative  position  of 
the  nebulas  in  the  interval  from  1785  to  1862,  which  seem  to 
indicate  a  motion  of  revolution  of  the  one  around  the  other. 

466*  Variability  of  Nebulae.  Systematic  observations 
have  been  made  by  Struve,  D'Arrest,  and  other  astronomers, 
with  the  view  of  ascertaining  whether  any  of  the  nebulge  were 
subject  to  variations  of  brightness.  The  result  is  that  in  a  small 
number  of  cases  some  degree  of  variability  has  been  positively 
ascertained.  One  case  is  that  of  the  nebula  in  Orion,  in  certain 
parts  of  which  material  changes  of  brightness  have  been  observed. 
But  the  most  marked  case  is  that  of  a  small  and  faint  nebula, 
discovered  by  Hind,  in  1852,  in  the  constellation  Taurus.  It 
has  since  gradually  faded  from  year  to  year,  and  in  1862  was 
barely  discernible  in  the  great  Pulkowa  refractor.  It  is  now 
entirely  invisible  in  the  telescope  with  which  it  was  first  detect- 
ed. It  is  an  interesting  fact  that  this  diminution  of  brightness 
has  proceeded  pari  passu  with  that  of  a  small  star  which  pre- 
sented itself  almost  in  contact  with  the  nebula.  It  has  been 
observed  also  that  there  are  many  variable  stars  in  a  part  of  the 
nebula  in  Orion  that  is  subject  to  change.  Corresponding 
changes  have  been  observed  in  the  faint  nebulous  haze  noticed 
around  some  of  the  variable  stars ;  for  instance,  the  new  star 
that  suddenly  burst  forth  in  May,  1866,  in  Corona  Borealis,  and 
then  rapidly  declined  in  brightness. 


0 


DISTANCE  AND  MAGNITUDE  OF  NEBULJE.  267 


DISTANCE  AND  MAGNITUDE  OF  NEBULAE. 

467.  Resolved  Nebulae.  Herschel  undertook  to  determine 
the  distance  of  resolved  nebulae,  by  noting  the  space-penetrating 
power  of  the  telescope  which  first  succeeded  in  revealing  their 
distinct  stars.  According  to  his  determinations,  therefore,  the 
most  remote  of  the  resolved  nebula?  are  at  the  same  distance  as 
the  most  remote  of  the  isolated  stars  discerned  in  his  large 
telescope.  The  theoretical  space-penetrating  power  of  his  tele- 
scope was  2,080  times  the  mean  distance  of  stars  of  the  first 
magnitude.  This  should  accordingly  be  the  limiting  distance  of 


the  resolved  nebulae  seen  in  Herschel's  telescope.  The  corres- 
ponding limit  for  stars  and  nebulae,  as  seen  in  Lord  Rosse's  tele- 
scope, should  be  3,120.  But  Struve,  after  determining  the  com- 
parative distances  of  stars  of  the  different  photometric  magni- 
tudes, by  comparing  the  actual  number  of  stars  of  the  different 
magnitudes,  has  been  enabled  to  ascertain  the  actual  space-pen- 
etrating power  of  any  telescope  in  which  all  the  stars  up  to  any 
particular  magnitude  could  be  seen.  According  to  his  deter- 
minations, the  actual  space-penetrating  power  of  Herschers  tele- 
scope of  20  feet  focus  was  183;  that  of  the  40  feet  reflector 
was  368,  instead  of  2080  as  deduced  upon  optical  principles ;  and 
that  of  Lord  Rosse's  great  telescope  is  422,  instead  of  3,120,  the 
theoretical  determination. 

The  unit  of  distance  in  these  numerical  values  is  the  mean 
distance  of  stars  of  the  first  magnitude.  According  to  Peters, 
this  corresponds  to  a  parallax  of  0".*Jl,  and  is  traversed  by  light 
in  15.5  years.  We  may  therefore  conclude  that  light  employs 
about  6,540  years  in  coming  from  the  most  remote  telescopic 
stars  hitherto  discerned  to  the  earth.  It  traverses  the  distance 
from  the  nearest  star  (a  Centauri)  to  the  earth  in  3J  years. 

The  resolvable  nebulae  require  telescopes  of  various  powers 
to  reveal  their  individual  stars,  and  must  therefore  be  distributed 
at  the  same  variety  of  distance  as  the  isolated  telescopic  stars 
of  similar  magnitudes. 

46§.  Irresolvable  Nebulae.  Herschel  also  undertook  to 
determine  the  probable  distance  of  the  more  remote  irresolvable 
nebulae.  He  estimated  that  a  certain  cluster  of  stars  (75  of  Mes- 
sier's  catalogue),  which  at  one-fourth  of  its  distance  would  be 
visible  to  the  naked  eye,  would  be  visible  as  a  faint  irresolvable 
nebula,  in  his  great  reflector,  if  it  were  removed  to  48  times  its 
actual  distance,  or  to  more  than  35,000  times  the  distance  of 
Sirius.  Struve's  investigation  reduces  this  determination  to  787 
times  the  mean  distance  of  stars  of  the  first  magnitude  (467). 
The  corresponding  result  for  Lord  Rosse's  telescope  would  be 
only  a  small  fraction  greater. 

469.  Extinction  of  the  Light  of  the  Stars,  in  its  passage 


268  THE  FIXED  STARS. 

through  space.  The  course  of  investigation  followed  up  by 
Struve,  at  the  same  time  that  it  affixed  a  much  lower  limit  to 
the  power  of  telescopes  to  pierce  into  the  depths  of  space,  con- 
ducted in  explanation  of  this  fact,  to  an  important  theoretical 
conclusion,  viz..  that  the  light  of  the  stars  is  partially  extinguished 
in  its  transit  through  space.  He  estimated  the  amount  of  this 
extinction  to  be  such  that  light,  in  its  passage  through  a  distance 
equal  to  that  of  a  star  of  the  first  magnitude,  loses  y^-  of  its  intens- 
ity. Sir  John  Herschel  controverts  this  theory  of  the  distinguished 
Pulkowa  astronomer,  but  makes  no  attempt  to  overthrow  the 
principal  argument  upon  which  it  rests.  If  we  reject,  with  Her- 
schel, the  testimony  of  the  stars  relative  to  the  power  of  telescopes 
to  penetrate  the  depths  of  space  in  which  they  lie,  we  must  then 
adopt  the  determinations  obtained  upon  optical  principles  alone 
as  the  exponents  of  telescopic  power;  we  must  accordingly  con- 
clude that  stars  can  be  discerned  with  the  most  powerful  tele- 
scopes when  separated  from  us  by  a  distance  so  vast  that  light 
requires  48,000  years  to  traverse  it;  and  that  nebulae  might  still 
be  visible  at  a  distance  which  light  would  require  500,000  years 
to  pass  over.  At  that  distance,  the  united  impression  of  the 
light  of  10,000  stars  upon  the  eye  would  only  equal  that  from 
100  single  stars,  so  remote  as  to  be  just  discernible  in  the  most 
powerful  telescope ;  and  therefore  clusters  containing  hundreds 
of  thousands  of  stars  should  be  visible  at  a  much  greater  dis- 
tance. 

4TO.  Magnitude  of  JVelmlae.  At  the  distance  of  422 
stellar  intervals  (the  utmost  actual  reach  of  Lord  Kosse's  teles- 
cope) a  linear  extent  of  10',  in  the  heavens,  answers  to  1.23 
times  one  of  these  intervals  (467).  Some  of  the  planetary  ne- 
bulae have  an  apparent  diameter  as  great  as  10',  and  as  they 
are  probably  more  remote  than  the  most  distant  telescopic  stars, 
their  actual  diameters  are  probably  greater  than  1.23  stellar 
units.  The  irregular  nebulae  have  a  much  greater  extent.  For 
example,  the  more  conspicuous  portion  of  the  nebula  in  Orion 
extends  to  30',  or  3.7  stellar  intervals,  in  the  east  and  west  direc- 
tion, and  nearly  as  far  in  the  north  and  south  direction.  The 
outlying  branches  run  out  much  further.  The  extreme  length 
of  the  nebula  in  Andromeda  is  no  less  than  18  times  the  same 
•unit  or  the  mean  distance  of  stars  of  the  first  magnitude.  Its 
extreme  breadth  is  7-J-  units.  We  here  suppose  these  two  nebulae 
to  be  at  the  distance  of  the  most  remote  telescopic  stars.  As 
they  are  barely  resolvable  by  the  most  powerful  telescopes,  their 
distance  cannot  be  less  than  this,  unless  their  component  stars 
are  smaller,  or  intrinsically  less  luminous  than  the  more  remote 
isolated  stars. 

If  the  space-penetrating  power  of  telescopes,  as  obtained  upon 
optical  principles,  be  adopted,  the  above  numerical  results  must 
be  increased  seven- fold. 


COMPONENT  STARS  OF  CLUSTERS.  269 


NUMBER,  MUTUAL  DISTANCE,  AND  COMPARATIYE  BRIGHT- 
NESS OF  THE  COMPONENT  STARS  OF  CLUSTERS. 

471.     Po§§ible    Number  of    Star*    in    a    Nebula.       We 

may  obtain  an  approximate  estimate  of  the  number  of  stars  that 
may  be  congregated  together  in  a  nebula  that  is  completely  re- 
solvable by  a  powerful  telescope,  by  considering  that  if  the  tel- 
escope just  shows  them  distinctly  separate,  the  apparent  distance 
between  two  contiguous  stars  may  be  assumed  to  be  less  than  V '. 
A  space  of  one  square  minute  should  then  contain  more  than 
3,600  stars.  The  planetary  nebula  near  the  star  p,  in  the  con- 
stellation of  the  Great  Bear  (461),  has  an  apparent  extent  of  7 
square  minutes.  If  it  were  just  resolvable  it  should  then  con- 
tain more  than  25,000  stars.  As  it  is  really  irresolvable,  the 
number  of  its  individual  stars  must  be  still  greater.  Upon  the 
surne  basis  of  calculation,  the  more  conspicuous  portion  of  the 
nebula  in  Orion,  occupying,  according  to  Sir  John  Herschel,  -^ 
of  a  square  degree,  should  contain  more  than  500,000  stars ;  and 
the  similar  portion  of  the  nebula  in  Andromeda  (90'  long  by 
15'  broad)  not  less  than  4,000,000  stars.  If  we  suppose  this 
vast  nebula  to  be  one  continuous  bed  of  stars,  of  different  sizes, 
for  its  entire  extent,  it  must  comprise  the  enormous  number  of 
30,000,000  stars. 

It  is  true  that  these  great  nebulae  where  resolved,  in  their 
brighter  portions,  show  distinct  stars  in  numbers  that  can  be 
counted ;  but  the  space  intervening  between  them  is  full  of  a 
nebulosity  that  is  probably  composed  of  smaller  stars  too  closely 
compacted  to  be  separated  by  tne  telescope. 

4T2.  Limit  of  Distance  between  Star*  in  a  Resolved 
Nebula.  An  angular  space  of  1",  at  a  distance  equal  to  422 
stellar  intervals,  corresponds  to  a  linear  distance  2,019  times  the 
distance  of  the  earth  from  the  sun,  or  about  67  times  the  radius 
of  Neptune's  orbit.  The  distance  between  two  contiguous  stars 
of  a  nebula,  that  are  just  separated  by  a  powerful  telescope,  can- 
not exceed  this  amount. 

If  the  light  of  the  stars  suffers  no  sensible  extinction  in  its 
passage,  and  therefore  telescopes  really  penetrate  as  far  into  space 
as  the  optical  theory  requires,  this  determination  is  only  ^  of  the 
actual  value. 

Clusters  whose  individual  stars  are  separated  by  the  distance 
just  determined,  would,  if  posited  at  a  less  distance  than  the 
furthest  reach  of  telescopes,  be  more  readily  resolved  ;  while  any 
that  might  be  at  a  greater  distance  would  be  wholly  irresolvable 
by  any  telescope  yet.  constructed. 

473.  Explanation  of  Inequalities  of  Brightness  in  a 
Nebula.  Globular  and  irregular  clusters  (456-7,)  are  brighter 
and  more  difficult  of  resolution  at  the  central  than  at  the  outer 


270  THE   FIXED  STARS. 

portions  of  the  cluster.  This  is  what  should  result  if  they  were 
composed  of  stars  of  equal  size  and  equally  spaced.  But  in 
some  instances  the  increase  of  brightness  towards  the  centre  is  too 
great  to  admit  of  this  supposition ;  and  we  infer  that  the  stars 
are  there  condensed  into  a  smaller  space. 

Oval  and  irregular  nebulae  are  more  difficult  of  resolution  at 
the  fainter  than  at  the  brighter  parts.  From  this  we  may  infer 
that  the  stars  are  larger  or  more  luminous  in  the  brightest  por- 
tions of  such  nebulae;  or  that  instances  of  close  juxtaposition 
more  frequently  occur,  in  groups  of  two  or  three,  which  appear 
united  as  one,  as  suggested  by  Sir  John  Herschel. 


STRUCTURE  OF  THE  SIDEREAL  UNIVERSE. 

474.  Sy§tem  of  the  Milky  Way.  We  have  already  seen 
(428)  that  Sir  William  Herschel  made  the  grand  discovery  that 
the  sun  is  one  of  the  individual  stars  of  a  vast  bed,  or  organized 
system  of  stars,  called  the  system  of  the  milky  way ;  that  the 
sun  is  posited  near  its  middle  plane,  and  that  its  innumerable 
stars  constitute  the  starry  host  which  diversify  our  firmament. 
He  at  first  conceived  that  his  telescope  penetrated  to  the  outer- 
most limits  of  the  stratum,  but  later  investigations,  recently  con- 
firmed by  the  observations  and  researches  of  Bessel,  Argelander, 
and  Struve,  have  fully  established  that  it  extends  in  all  directions 
beyond  the  reach  of  the  most  powerful  telescopes  ;  and  that  we 
can  obtain  no  definite  knowledge  of  its  exterior  form. 

Herschel's  star-gauges  afford  positive  information  only  with 
regard  to  the  comparative  densities  of  the  fathomless  starry  stra- 
tum in  different  directions,  within  the  range  of  telescopic  vision. 
From  these  we  learn  that  the  individual  stars  are  not  uniformly 
distributed  throughout  the  system,  but  are  greatly  condensed 
towards  the  medial  plane.  Struve,  by  an  elaborate  discussion, 
has  established  that  the  distance  between  neighboring  stars  de- 
creases, according  to  a  regular  law,  on  both  sides  of  this  plane  as 
the  distance  from  it  increases;  the  decrease  being  much  more 
rapid  at  first,  and  the  rate  gradually  declining  with  the  increas- 
ing distance.  Within  this  plane  of  greatest  condensation  there 
is  also  a  line  of  greatest  density,  from  both  sides  of  which  the 
density  gradually  decreases.  A  corresponding  line  of  superior 
density  exists  in  each  plane  of  the  starry  stratum  parallel  to  the 
principal  plane.  The  axis  of  greatest  condensation  is  nearly 
coincident  with  the  line  passing  through  the  points  of  intersec- 
tion of  the  galactic  circle,  or  middle  line  of  the  milky  way  in 
the  heavens,  with  the  equator.  These  points  lie  in  K.  Asc.  6h. 
40m.,  and  E.  Asc.  18h.  40m.,  between  the  constellations  Orion 
and  Canis  Minor,  and  between  Serpentarius  and  Antinous.  Ac- 
cording to  Struve  the  sun  is  on  the  north  side  of  the  plane  of  great- 


STRUCTURE   OF  THE  SIDEREAL   UNIVERSE.  271 

est  condensation,  and  at  an  estimated  distance  from  it  equal  to 
the  distance  of  *  Centauri  from  the  sun  and  earth.  It  is  also  to 
one  side  of  the  axis  of  greatest  density  in  the  direction  of  the 
constellation  Virgo,  and  at  a  distance  nearly  equal  to  the  distance 
of  the  nearest  stars  of  the  second  magnitude  from  the  earth. 
The  galactic  circle,  and  therefore,  also,  the  principal  plane  of  the 
milky  way,  passes  through  the  points  on  the  equator  above-men- 
tioned, and  within  about  30°  of  the  north  and  south  poles  of  the 
heavens ;  through  points  in  the  constellations  Cassiopeia  and  the 
Southern  Cross.  The  north  pole  of  the  galactic  circle,  or  of  the 
whole  system,  lies  in  K.  Asc.  12h.  38m.,  and  Dec.  31°.5,  between 
the  constellations  Coma  Berenices  and  Canes  Yenatici. 

475.  The  ttalaxy,  or  Belt  of  the  Milky  Way.     The  lu- 
minous belt  in  the  heavens  called  the  milky  way,  as  seen  by  the 
naked  eye,   varies  in  breadth  at  different  points  between  the 
limits  5°  and  16°,  and  has  an  average  breadth  of  about  10°.     It 
presents  a  succession  of  luminous  patches,  unequally  condensed, 
intermingled  with  others  of  a  fainter  shade.     From  the  bright 
star  *  Cygni,  in  the  northern  hemisphere,  it  runs  towards  the 
southwest  in  two  clustering  streams,  which  reunite  beyond  the 
southern  constellation  Scorpio,  at  a  distance  of  120°  from  the 
point  of  separation.     Near  the  place  in  which  it  crosses  the  equa- 
tor, between   Antinous  and   Serpentarius,    the   double   stream 
attains  its  greatest  width  of  22°.     The  middle  point  of  crossing 
is  the  ascending  node,  on  the  equator,  of  the  galactic  circle. 

To  give  a  more  accurate  idea  of  the  S3rstem  of  the  milky  way, 
we  must  add  that  its  principal  plane,  so  called,  is  not  strictly  a 
single  plane,  but  a  broken  plane,  or  two  planes  differing  about 
10°  in  their  direction,  and  separating  at  the  line  of  the  nodes  in 
the  equator.  The  two  condensed  branches  answering  to  the  two 
separate  streams  in  the  heavens  just  noticed,  lie  on  opposite  sides 
of  this  broken  plane.  The  line  of  greatest  density  before  referred 
to  (474)  also  is  not  truly  a  right  line,  but  has  sensible  inflexions; 
and  there  occur  in  its  vicinity  remarkable  alternations  of  starry 
condensations  and  vacant  spaces.  Similar  interruptions  of  con- 
tinuity are  observed  in  various  directions  through  the  mass.  In 
some  directions  dark  intervening  spaces  are  seen,  in  which, 
according  to  Sir  John  Herschel,  the  telescope  seems  to  penetrate 
to  the  very  confines  of  the  starry  stratum.  In  other  directions, 
there  appear  to  be  vast  starless  regions  lying  between  the  more 
remote  portions  and  outlying  branches  of  the  milky  way,  or  other 
systems  entirely  detached  from  it. 

476.  Relations  of  Clutters  and  tfebnlae  to  the  System 
of  the  milky  Way.     Globular  and  irregular  clusters  are  far 
more  abundant  in  the  denser  portions  of  the  milky  way  than  in 
other  portions  of  equal  extent.     The  irregular  nebula?,  some  of 
which  have  been  resolved,  are,  for  the  most  part,  either  portions 
or  outlying  branches  of  the  system.     Some  of  those  which  have 


272  THE   FIXED  STARS. 

not  been  resolved,  may  possibly  be  independent  systems  exterior 
to  that  of  the  milky  way. 

Oval  nebulse,  and  the  irresolvable  nebulae  generally,  do  not 
hold  the  same  relations  to  our  starry  firmament.  They  are 
mostly  absent  from  that  great  belt  in  which  the  stars  are  so 
numerous  and  condensed,  and  the  conspicuous  clusters  abound, 
and  are  congregated  towards  its  poles.  The  region  richest  in 
nebulae  lies  around  its  north  pole.  They  are  more  uniformly 
disseminated  and  more  widely  dispersed  over  the  zone  which 
surrounds  its  south  pole ;  and  are  at  the  same  time  less  numer- 
ous. But  on  the  other  hand,  as  already  intimated,  there  are  two 
luminous  tracts  of  the  southern  heavens,  called  the  Magellanic 
Clouds,  in  which  they  occur  in  large  numbers.  In  these  they 
are  found  associated  with  groups  and  clusters  of  stars  of  every 
form,  and  must  be  presumed  to  be  no  more  remote  than  these 
resolved  clusters.  In  the  northern  hemisphere  they  in  general 
occur  dissociated  from  resolved  clusters,  and  may  be  much  more 
remote.  According  to  the  estimate  already  obtained  (468)  their 
extreme  limit  of  distance  does  not  exceed  twice  that  of  the  most 
distant  isolated  stars  visible  in  telescopes. 

477.  Theoretical  Inferences.  The  peculiarity  that  has  just  been  noticed 
in  the  position  of  most  of  the  oval  and  irresolvable  nebulae  of  the  northern  hemi- 
sphere, leads  to  the  supposition  that  they  may  have  originated  in  a  different  manner 
from  the  clusters  and  nebulae  that  are  chiefly  accumulated  in  the  denser  portions 
of  the  system  of  the  milky  way,  and  undoubtedly  are  component  parts  of  it;  and 
that  they  may  differ  from  these  in  some  of  the  features  of  their  physical  constitu- 
tion. The  latter  supposition  acquires  additional  probability  from  a  recent  discovery 
that  the  character  of  the  light  received  from  some  of  the  nebulae  is  in  certain  re- 
spects different  from  that  of  the  light  received  from  the  sun  and  the  stars.  A 
spectral  analysis  of  the  light  from  some  of  these  nebulae,  by  two  eminent  physi- 
cists, has  disclosed  the  remarkable  fact,  that  it  is  not  made  up  of  rays  of  widely 
different  refrangibilities,  but  is,  the  greater  part  of  it,  monochromatic ;  and  that 
the  spectrum  is  not  crossed  by  dark  lines,  like  that  obtained  from  the  light  of  the 
sun,  or  of  a  star.  From  this,  the  experimenters  draw  the  conclusion  that  the  ne- 
bulas in  question  can  no  longer  be  regarded  as  clusters  of  suns,  similar  in  constitu- 
tion to  the  centre  of  our  planetary  system,  but  as  objects  having  quite  a  different 
and  peculiar  composition ;  and  that  instead  of  being  considered  as  made  up  of 
bodies  having  a  solid  nucleus,  they  must  be  regarded  as  enormous  masses  of  lumi- 
nous gas  or  vapor.  The  latter  conclusion  does  not  follow  of  necessity  from  the 
results  of  the  experiments ;  they  only  show  that  the  light  from  these  nebulae  comes 
from  masses  of  pure  gas  or  vapor,  rendered  luminous  either  by  ignition  or  electric 
discharges,  but  afford  no  certain  knowledge  with  regard  to  the  existence  of  a  solid 
nucleus. 

47§.  General    Motion    of    Revolution    of    the    Stars. 

Madler,  after  an  elaborate  discussion  of  the  proper  motions 
of  a  large  number  of  stars,  has  arrived  at  the  conclusion 
that  the  collective  body  of  stars  visible  to  us  has,  together 
with  the  sun,  a  common  movement  of  revolution  around  a  cen- 
tre situated  in  the  group  of  the  Pleiades.  He  estimates  the 
period  of  revolution  to  be  about  27  millions  of  years.  A 
general  circulation  of  the  sun  and  the  stars  of  our  firmament 
around  a  common  centre  of  attraction,  must  also  be  regarded  as 
highly  probable  upon  physical  grounds,  but  it  cannot  be  doubted 


DYNAMICAL   CONDITION  OF  SIDEREAL  SYSTEMS.  273 

that  the  centre  of  attraction  would  lie  in  the  principal  plane  of 
the  milky  way.  The  group  of  the  Pleiades  lies  considerably  to 
the  south  of  this  plane,  and  therefore  in  all  probability  the  actual 
centre  is  situated  to  the  north  of  the  Pleiades,  in  the  constella- 
tion Perseus,  as  suggested  by  Argelander. 

479.  Hypotheses  respecting  the  Milky  Way.  Madler 
supposes  that  the  stars  of  the  milky  way  are  arranged  in  seve- 
ral concentric  rings  of  unequal  thickness,  and  of  varying  dimen- 
sions in  different  directions,  but  lying  nearly  in  the  same  plane. 
He  conceives  the  sun  to  be  eccentrically  situated  in  the  sys- 
tem, and  at  a  short  distance  from  the  general  plane,  of  the  rings ; 
so  that  on  one  side  the  rings  are  seen  distinctly  separate. 

Professor  Stephen  Alexander,  of  Princeton  College,  has 
advanced  the  hypothesis  that  the  milky  way,  and  the  stars  within 
it,  together  constitute  a  spiral  with  several  branches,  and  a  cen- 
tral spheroidal  cluster. 

The  hypothesis  of  Sir  William  Herschel  has  already  been 
considered  (428  and  474).  Another  conception  of  the  probable 
structure,  and  present  dynamical  condition  of  the  system  of  the 
milky  way,  is  briefly  presented  in  a  Note  in  the  Appendix. 


GENERAL  DYNAMICAL  CONDITION   OF  SIDEREAL  SYSTEMS. 

4§O,  Three  different  general  conceptions  may  be  formed  of 
the  possible  nature  of  the  motions  of  the  individual  members 
of  a  cluster  or  system  of  stars. 

(1.)  They  may  all  be  in  the  act  of  falling  in  right  lines  towards 
their  common  centre  of  attraction. 

(2.)  They  may  be  in  the  act  of  receding  from  a  centre  about 
which  they  were  originally  collected,  under  the  influence  of  some 
dispersing  force. 

(3.)  They  may  be  revolving  in  separate  orbits  around  their 
common  centre  of  attraction,  or  possibly  around  different 
centres. 

First  Hypothesis. — This  was  proposed  by  Sir  William  Her- 
schel. It  accords  with  the  different  aspects  presented  by  clus- 
ters condensed  towards  a  centre,  but  cannot  be  applied  to  annular 
nebulae,  some  of  which  are  known  to  consist  of  stars,  nor  to 
spiral  formed  clusters.  It  involves  also  the  highly  improbable 
supposition  that  there  is  in  the  condition  of  the  system  no  provi- 
sion for  stability,  but  only  for  its  inevitable  destruction,  in  the 
final  collision  of  all  its  constituent  stars  at  its  centre. 

Second  Hypothesis. — The  second  supposition  is  advocated  by 
Professor  Alexander,  who  has  propounded  a  systematic  theory 
of  the  evolution  of  sidereal  systems,  under  the  operation  of  a 
certain  supposed  process  of  dispersion. 

Third  Hypothesis. — The  supposition  that  the  individual  stars 

18 


274  THE  FIXED  STARS. 

of  a  system  are  moving  in  separate  orbits  about  a  common  cen- 
tre of  attraction,  is  that  which  is  suggested  by  the  analogy  of 
our  planetary  system,  as  well  as  that  of  the  revolution  of  binary 
and  triple  stars  around  their  common  centre  of  gravity.  It  is 
supported  also  by  the  results  of  Madler's  investigations  with 
respect  to  a  general  revolution  of  the  system  of  the  milky  way 
about  a  centre  (478).  It  implies  the  existence  of  the  only 
causes  of  stability  that  can  be  conceived  to  be  in  operation  ;  viz., 
a  centre  of  attraction,  and  a  motion  of  revolution  around  that 
centre.  For  the  rotation  of  a  cluster  of  separate  stars  around 
an  axis,  as  one  single  body  of  matter,  is  mechanically  impossi- 
ble. In  the  history  of  such  an  organized  system,  from  its  be- 
ginning, there  may  be  epochs  of  collision  among  its  individual 
members,  but  when  all  such  cases,  inevitably  resulting  from  cor- 
respondences of  original  position,  have  occurred,  the  motions 
which  remain  outstanding  may  ultimately  tend  to  a  permanent 
stability. 


NEBULAR  HYPOTHESIS.  275 


CHAPTER  XX. 

THEORIES  OF  THE  EVOLUTION  OF  SIDEREAL  AND  PLANETARY 

SYSTEMS. 

NEBULAR  HYPOTHESIS. 

481.  Primitive  Nebulous  Condition  of  all  Systems.  Although  the 
telescope,  by  revealing  the  stellar  constitution  of  many  of  the  nebulae  regarded  by 
Sir  William  Herschel  as  giving  no  intimations  of  resolvability,  has  removed  the 
supposed  direct  evidence  of  the  existence  of  detached  masses  of  nebulous  matter 
disseminated  through  space,  there  still  remains  strong  indirect  evidence  of  a  pri- 
mitive nebulous  condition  of  all  worlds  and  systems  of  worlds.  Numerous  correspon- 
dences of  structural  and  dynamical  features,  and  intimations  of  a  progressive  crea- 
tion, lead  to  this  conception  as  the  only  ground  upon  which  they  can  reasonably  be 
explained.  Thus  Laplace  adduces  five  general  phenomona  as  indications  of  a  com- 
mon origin  of  the  system  of  planets  circulating  around  the  sun ;  and  infers  that 
they  must  all  have  originally  formed  portions  of  one  vast  nebulous  body  rotating 
about  an  axis.  These  are : 

1.  The  planets  all  revolve  in  the  same  direction  around  the  sun ;  viz. :  from  west 
to  east. 

2.  Their  orbits  lie  nearly  in  the  plane  of  the  sun's  equator. 

3.  Their  orbits  are  ellipses  of  small  eccentricity. 

4.  The  sun  and  all  the  planets,  so  far  as  the  circumstances  of  their  rotation  are 
known,  rotate  about  axes  in  the  same  direction  that  the  planets  revolve  around 
the  sun. 

5.  The  satellites  revolve  around  their  primaries  in  the  same  direction  that  these 
revolve  around  the  sun,  and  turn  about  their  axes.    They  also  revolve,  as  far  as 
known,  approximately  in  the  plane  of  the  equator  of  each  primary ;  and  describe 
ellipses  of  small  eccentricity. 

The  only  known  exception  to  the  general  direction  of  revolution  occurs  in  the 
case  of  the  satellites  of  Uranus,  which  have  a  common  retrograde  motion.  Their 
orbits  are  also  inclined  to  the  plane  of  the  ecliptic  under  a  large  angle  (79°) ;  but 
their  common  plane  may  still  coincide  with  the  plane  of  Uranus's  equator,  and  the 
direction  of  their  motion  of  revolution  may  be  the  same  as  that  of  the  rotation  of 
the  primary.  (See  Note  III.). 

The  hypothesis  proposed  by  Herschel  in  explanation  of  sidereal  systems,  and 
since  extended  by  Laplace  to  the  explanation  of  the  solar  system,  is  called  the 
Nebular  Hypothesis.  It  is,  comprehensively  stated,  that  all  worlds  and  systems  of 
worlds  have  been  slowly  evolved  from  primordial  nebulous  masses,  under  the 
operation  of  the  general  forces  and  properties  which  the  Creator  has  either  per- 
manently imparted  to  matter,  or  is  incessantly  renewing  in  it 


DEVELOPMENT  OF  THE  SOLAR  SYSTEM. 

482.  Origin  of  the  Planets  and  Satellites.  The  mechanical  theory  of 
the  formation  of  the  solar  system  propounded  by  Laplace,  is  briefly  this :  The 
rotating  nebulous  body  from  which  the  system  has  been  evolved,  in  the  progress 
of  ages  slowly  contracted  and  condensed,  by  the  gravitation  of  its  parts  towards  the 
centre,  and  by  the  process  of  cooling  at  its  surface.  This  contraction  of  necessity 
accelerated  the  rotation  of  the  body,  and  augmented  the  centrifugal  force :  until 


2T6     EVOLUTION  OF  SIDEREAL  AND   PLANETARY  SYSTEMS. 

finally  the  increasing  centrifugal  force  at  the  equator  balanced  the  gravity.  "When 
this  mechanical  condition  was  reached  at  the  surface,  and  for  a  certain  depth 
where  the  influence  of  the  cooling  had  especially  prevailed,  a  vaporous  zone  became 
detached,  and  revolved  independently  of  the  interior  mass.  This  zone,  by  concen- 
tration at  special  points,  eventually  separated  into  fragments ;  which,  from  the 
preponderating  attraction  of  the  larger  fragment,  or  because  of  slight  differences 
of  initial  velocity,  became  incorporated  into  one  revolving  body.  This  body  would 
take  up  a  motion  of  rotation  in  the  same  direction  that  it  revolves ;  since  the  parts 
most  remote  from  the  sun  would  have  the  most  rapid  motion  of  revolution.  By 
an  indefinite  continuation  of  the  same  process  a  succession  of  zones  would  become 
detached,  and  a  system  of  vaporous  bodies  revolving  around  a  central  condensed 
mass  would  be  formed.  Each  of  these  revolving  bodies  being  also  in  the  same 
condition  of  rotation  as  the  original  nebulous  mass,  might  pass  through  a  similar 
succession  of  changes,  and  thus  a  system  of  satellites  circulating  around  a  primary, 
in  the  direction  of  the  rotation,  be  developed. 

The  solar  system  presents  one  instance,  that  of  Saturn's  ring,  in  which  the  de- 
tached vaporous  zone  condensed  uniformly  without  separating  into  parts.  The 
planetoids  appear  to  afford  an  instance  of  the  opposite  extreme,  in  which  the 
ring  broke  up  into  a  great  number  of  small  fragments  that  continued  to  revolve 
separately. 

483.  Origin  of  Cometary  Bodies.  Laplace  supposed  the  comets  did  not 
belong,  originally,  to  the  solar  system,  but  wandered  into  its  precincts  from  other 
systems,  and  so  became  permanently  united  with  it  by  the  bond  of  gravita- 
tion. But,  with  the  evidence  now  afforded  by  accumulated  facts,  several  con- 
siderations may  be  urged  which  tend  to  show  that  comets  have  been  derived  from 
the  same  nebulous  body  as  the  planets  and  satellites.  The  principal  of  these  are 
the  following: 

1.  The  comets  of  short  period  form  a  class  but  little  distinguished,  in  their  orbit- 
al motions,  from  the  planetoids.     They  revolve  in  the  same  direction,  and  in  orbits 
having  about  the  same  average  inclination  to  the  ecliptic,  as  those  of  the  planetoids. 
Their  orbits  are  only  somewhat  more  eccentric. 

2.  All  the  known  comets  that  describe  orbits  whose  aphelia  lie  within  the  limits 
of  the  solar  system,  or  do  not  fall  more  than  fifty  millions  of  miles  beyond  the  orbit 
of  Neptune,  revolve  in  the  same  direction  as  the  planets. 

3.  If  we  compare  all  the  comets  whose  elliptic  orbits  have  been  determined  with 
more  or  less  accuracy,  among  themselves,  we  find  that  the  more  eccentric  orbits  of 
the  comets  of  long  period  are  more  inclined  to  the  plane  of  the  ecliptic  than  the 
less  eccentric  orbits  of  the  comets  of  short  period. 

If  we  consider  the  class  of  comets  which  recede  to  a  distance  of  more  than  fifty 
millions  of  miles  beyond  the  limits  of  the  solar  system,  it  appears  that  as  many 
among  them  have  a  retrograde  as  a  direct  motion ;  while  the  majority  move  in 
orbits  inclined  under  large  angles  to  the  ecliptic.  These  exceptional  facts  do  not 
necessarily  imply  that  this  class  of  comets  have  an  origin  extraneous  to  the  sys- 
tem ;  but  rather  that  the  mode  of  their  evolution  from  the  primary  nebulous  body 
was  different  from  that  of  the  planets  and  comets  of  short  period.  Now,  besides 
the  process  of  evolution  supposed  to  have  been  in  operation  in  the  case  of  the 
planets,  we  may  conceive, 

(1.)  That  certain  portions  of  the  body,  near  its  surface,  became,  by  mutual  attrac- 
tion of  their  parts  and  by  cooling,  condensed  upon  particular  points  into  masses 
of  sufficient  density  to  revolve  independently.  Such  masses,  as  they  would  have 
less  initial  velocities  in  proportion  as  they  were  more  remote  from  the  equator, 
would,  in  general,  describe  orbits  more  eccentric  in  proportion  as  they  are  more 
inclined  to  the  ecliptic.  Besides,  the  masses  which  became  detached  at  the  equa- 
tor in  the  manner  here  supposed,  must  have  separated  from  the  general  mass  in 
the  intervals  between  the  epochs  of  the  separation  of  the  equatorial  planetary 
rings,  during  which  the  velocity  of  rotation  at  the  equator  was  less  than  that  an- 
swering to  a  motion  of  revolution  in  a  circle.  The  comets  of  the  first  two  classes 
may  have  thus  originated.  If  so,  as  they  must  have  performed  many  revolutions 
within  the  attenuated  mass  of  the  nebulous  body,  they  are  now  doubtless  moving 
in  orbits  much  more  eccentric  than  those  which  they  first  described. 

(2.)  That  fragments  may  have  been  suddenly  detached  from  the  general  nebu- 
lous mass,  by  the  operation  of  some  expelling  force.  If  we  adopt  the  most  prob- 
able hypothesis,  that  this  force  acted  indifferently  in  all  directions  outward  from 


DEVELOPMENT  OF  THE  SOLAR  SYSTEM.  277 

the  surface,  and  assume  it  to  have  been  of  sufficient  intensity  to  impart,  when 
exerted  under  certain  obliquities  to  the  surface,  a  velocity  in  the  direction  of  the 
parallel  of  latitude  considerably  greater  than  the  velocity  of  rotation  at  the  place 
of  discharge,  then  among  the  comets  thus  originating  that  come  within  our  firma- 
ment, a  retrograde  may  be  as  frequent  as  a  direct  motion.  For,  those  which  were 
detached  with  the  higher  velocities,  either  obliquely  in  the  direction  of  the  rota- 
tion or  in  the  opposite  direction,  would  move  hi  too  large  orbits  to  become  visible 
from  the  earth.  If  all  the  comets  detached,  however,  could  be  seen,  there  should 
be  a  preponderance  in  the  number  of  those  having  a  direct  motion.  (See  Note 
III.  in  Appendix.) 


PAET  II. 


PHYSICAL   ASTRONOMY. 


CHAPTER  XXI. 


PRINCIPLE  OF  UNIVERSAL  GRAVITATION. 


484.  Force  of  Gravity.  IT  is  demonstrated  in  treatises  on 
Mechanics,  that  if  a  body  move  in  a  curve  in  such  a  manner 
that  the  areas  traced  by  the  radius- vector  about  a  fixed  point, 
increase  proportionally  to  the  times,  it  is  solicited  by  an  inces- 
sant force  constantly  directed  towards  this  point. 

The  following  is  a  geometrical  proof  of  this  principle.  Conceive  the  orbit  to  be 
a  polygon  of  an  infinite  number  of  sides.  Let  ABCD  (Fig.  109)  be  a  portion  of  it  ; 
and  S  the  fixed  point  about  which  the  radius- vector 
describes  areas  proportional  to  the  times,  or  equal 
areas  in  equal  times.  Since  the  impulses  are  only 
communicated  at  the  angular  points  A,  B,  C,  D, 
&c.,  of  the  polygon,  the  motion  will  be  uniform 
along  each  of  the  sides  AB,  BC,  CD,  &c. :  and  since 
we  may  suppose  the  times  of  describing  these  sides 
to  be  equal,  we  shall  have  the  triangular  area  SAB 
equal  to  the  triangular  area  SBC,  and  SBC  equal  to 
SCD,  &c.  Produce  AB  and  make  Be  equal  to  AB, 
which  may  be  taken  to  represent  the  velocity  along 
AB ;  and  join  Ce.  Cc  will  be  parallel  to  the  line  of 
direction  of  the  impulse  that  takes  effect  at  B.  Upon 
SB  let  fall  the  perpendiculars  Am,  en,  Gr.  Then, 
since  AB  =  Be,  Am  =  en ;  and  since  the  equivalent 
triangles  SAB,  SBC,  have  a  common  base  SB,  Am 
=  Gr.  It  follows,  therefore,  that  en  =  Gr,  and 
consequently,  that  Cc  is  parallel  to  BS.  The  im- 
pulse which  the  body  receives  at  B  is  therefore 

directed  from  B  towards  S.     In  the  same  manner  S 

it  may  be  shown  that  the  impulse  which  it  receives  -^      ,  rtQ 

at  C  is  directed  from  C  towards  S.      The  line  of 

direction  of  the  force  passes,  therefore,  in  every  position  of  the  body,  through  the 
point  S. 

Now,  by  Kepler's  first  law,  the  areas  described  by  the  radius- 
vectors  of  the  planets  about  the  sun,  are  proportional  to  the 
times.  It  follows  therefore  from  this  law,  that  each  planet  is 
acted  upon  by  a  force  which  urges  it  continually  towards  the 
sun. 


PRINCIPLE   OF   UNIVERSAL   GRAVITATION. 


279 


This  fact  is  technically  expressed  by  saying  that  the  planets 
gravitate  towards  the  sun,  and  the  force  which  urges  each  planet 
towards  the  sun  is  called  its  Gravity,  or  Force  of  Gravity,  towards 
the  sun. 

4§5.  Its  Law  of  Variation.  It  is  also  proved  by  the  prin- 
ciples of  Mechanics,  that  if  a  body,  continually  urged  by  a  force 
directed  to  some  point,  describe  an  ellipse  of  which  that  point 
is  a  focus,  the  force  by  which  it  is  urged  must  vary  inversely 
as  the  square  of  the  distance. 

Thus,  let  ABG  (Fig.  110)  be  the 
supposed  elliptic  orbit  of  the  body, 
CA  and  CB  its  semi-axes,  and  S  the 
focus  towards  wh'.ch  the  force  is  con- 
stantly directed.  Also  let  P  be  one 
position  of  the  body,  PR  a  tangent 
to  the  orbit  at  P ;  and  draw  RQ  par- 
rallel  to  PS,  Qw,  HI,  and  CD,  par- 
allel to  PR,  Qsc  perpendicular  to  SP, 
and  join  S  and  Q.  CP  and  CD  are 
semi-conjugate  diameters.  Denote 
them,  respectively,  by  A'  and  B' ; 
and  denote  the  semi-axes,  CA  and 
CB,  by  A  and  B.  Since  HI  is  par- 


B 


allel  to  PR,  and,  by  a  well-known 
property  of  the  ellipse,  the   angle 


FiG.  110. 

RPS  is  equal  to  the  angle  HPT,  PH  is  equal  to  PI :    and  since  HO  =  SC,  and  CB 
is  parallel  to  HI,  E  is  the  middle  of  SL     We  have,  therefore, 


_ 

Now  the  force  at  P  is  measured  by  2Pw  ;  and  we  may  state  the  proportion 

Pu  :  Pv  ::  PE  :  PC  ::  A  :  A'  ;  which  gives  Pv  =  Pu^L. 

jflL 

By  the  equation  of  the  ellipse  referred  to  its  centre  and  conjugate  diameters, 
Pa  and  DL, 


If  we  regard  Q  as  indefinitely  near  to  P,  then  Qu  =  Qv,  and  Gv  =  2CP  =  2A'  ; 
and  therefore 


But  Qt*  :  Qz  ::  PE  :  PF  ::  CA  :  PF: 

and,  by  Analytical  Geometry, 

CD  x  PF=CA  x  CB,  or,  CA  :  PF  ::  CD  :  CB  ::  B'  :  B. 

Hence        QM  :  Qx::B'  :  B,  Qti*  :  Qx*  ::B'«  :  B*,   and  Qti*=  Q^? 


Substituting  in  equation  (a),     Qa;*l—  =  -r-.2Pw;  whence  Qz  =r-—.2Po. 

B  A  A 

Qo;  _,      4te 

Now  triangular  area  SQP=:&=SP  x  —  ;  whence  Qz  =  =.      Substituting,  there 

2  SP'2 

results 

4ft*      B8  A  1 

=^=  —  .2Pw;  or  2Ptt  =  —  .4fcf  .=5.  ...  (I). 
SF2  B2         S? 


280  PRINCIPLE   OF   UNIVERSAL   GRAVITATION. 

To  compare  the  intensities  of  the  force  at  different  points  of  the  orbit,  we  must 
take  the  values  of  2Pw,  by  which  they  are  measured,  for  the  same  interval  of  time. 
On  this  supposition  k  is  constant,  and  therefore  the  force  is  inversely  proportional 
to  the  square  of  the  distance  SP. 

It  therefore  follows  from  Kepler's  second  law,  viz.  :  that  the 
planets  describe  ellipses  having  the  centre  of  the  sun  at  one  of 
their  foci  ;  that  the  force  of  gravity  of  each  planet  towards  the 
sun  varies  inversely  as  the  square  of  the  distance  from  the  sun's 
centre. 

4§6.  It  operates  on  all  the  Planets  alike.  By  taking 
into  view  Kepler's  third  law,  it  is  proved  that  it  is  one  and  the 
same  force,  modified  only  by  distance  from  the  sun,  which  causes 
all  the  planets  to  gravitate  towards  him,  and  retains  them  in  their 
orbits.  This  force  is  conceived  to  be  an  attraction  of  the  mat- 
ter of  the  sun  for  the  matter  of  the  planets,  and  is  called  the 
Solar  Attraction. 

To  deduce  this  consequence  from  Kepler's  third  law,  let  /,  t',  denote  the  perio- 
dic times  of  any  two  planets  ;  r,  r  ,  their  distances  from  the  sun  at  any  assumed 
point  of  time  ;  &,  k',  the  areas  described  by  their  radius-vectors  in  any  supposed 
unit  of  time;  and  A,  B,  and  A',  B',  the  semi-axes  of  their  elliptic  orbits.  Then  kt,  k't, 
will  be  equal  to  the  areas  of  the  entire  orbits  ;  which  are  also  measured  by  sAB, 

irA'B'. 

Thus  W:fcY::AB:A'B',and&«<»  :  &'2*'2  ::  A2B2  :  A'*B'«. 

But,  by  Kepler's  third  law,         I*  :  t'2  ::  A3  :  A'3. 

B2    B'2 

Dividing,  and  reducing,  k*  :  k'*  ::—  :  -~r  : 

Jo.      A 

that  is,  the  squares  of  the  areas  described  in  equal  times  are  as  the  parameters  of 
the  orbits. 

Now,  let  /,  /,  denote  the  forces  soliciting  the  two  planets.  Then,  by  equation 
(I),  Art.  485, 


From  which  it  appears  that  the  planets  are  solicited  by  a  force  of  gravitation 
towards  the  sun,  which  varies  from  one  planet  to  another  according  to  the  law  of 
the  inverse  square  of  their  distance. 

487.  Planets  Endued  with  an  Attractive  Foree.     The 

motions  of  the  satellites  are  in  conformity  with  Kepler's  laws  ; 
hence,  the  planets  which  have  satellites  are  endued  with  an  at- 
tractive force  of  the  same  nature  with  that  of  the  sun. 

The  existence  of  a  similar  attractive  power  in  each  of  the 
planets  that  are  devoid  of  satellites,  is  proved  by  the  fact  that 
the  observed  inequalities  of  their  motions,  and  of  those  of  the 
other  planets,  may  be  shown  upon  this  supposition  to  be  neces- 


NEWTON'S  THEORY  OF  UNIVERSAL  GRAVITATION.      281 

sary  consequences  of  the  attractions  of  the  planets  for  each 
other. 

In  like  manner  the  inequalities  in  the  motions  of  the  satellites 
and  their  primaries,  show  that  the  satellites  possess  the  same 
property  of  attraction  as  the  sun. 

4§§.  Tbe  Constituent  Particles  Attract  each  other.  We 
learn  from  the  motions  produced  by  the  action  of  the  sun  and 
planets  upon  each  other,  that  the  intensities  of  their  attractive 
forces  are,  at  the  same  distance,  proportional  to  their  masses,  and 
that  the  whole  attraction  of  the  same  body  for  different  bodies, 
is,  at  the  same  distance,  proportional  to  the  masses  of  these 
bodies.  From  which  we  may  infer  that  a  mutual  attraction 
exists  between  the  particles  of  bodies,  and  that  the  whole  force 
of  attraction  of  one  body  for  another,  is  the  result  of  the  attrac- 
tions of  its  individual  particles.  Moreover,  analysis  shows,  that 
in  order  that  the  law  of  attraction  of  the  whole  body  may  be 
that  of  the  inverse  ratio  of  the  square  of  the  distance,  this  must 
also  be  the  law  of  attraction  of  the  particles.  The  fact,  as  well 
as  the  law  of  the  mutual  attraction  of  particles,  is  also  revealed 
by  the  tides  and  other  phenomena  referable  to  such  attraction. 

4 §9.  Theory  of  Universal  Gravitation.  The  celestial 
phenomena  compared  with  the  general  laws  of  motion,  conduct 
us  therefore  to  this  great  principle  of  nature ;  namely,  that  all 
particles  of  matter  mutually  attract  each  other  in  the  direct  ratio  of 
their  masses^  and  in  the  inverse  ratio  of  the  squares  of  their  distances. 
This  is  called  the  principle  of  Universal  Gravitation.  The 
theory  of  its  existence  was  first  promulgated  by  Sir  Isaac  New- 
ton, and  is  hence  often  called  Newton's  Theory  of  Universal  Gra- 
vitation. The  force  which  urges  the  particles  of  matter  towards 
each  other  is  called  the  Force  of  Gravitation,  or  the  Attraction  of 
Gravitation. 

In  the  following  chapters  our  object  will  be  to  develop  the 
most  important  effects  of  the  principle  of  gravitation  thus  ar- 
rived at  by  induction.  The  perfect  accordance  that  will  be  ob- 
served to  obtain  between  the  deductions  from  the  theory  of 
universal  gravitation  and  the  results  of  observation,  will  afford 
additional  confirmation  of  the  truth  of  the  theory. 


THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 


CHAPTEE  XXII. 

THEORY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

49O.  Accelerating  Force  due  to  Smi'§  Attraction.    Let 

the  attraction  of  the  unit  of  mass  of  the  sun  for  the  unit  of 
mass  of  a  planet,  at  the  unit  of  distance,  be  designated  by  1. 
The  whole  attraction  exerted  by  the  sun  upon  the  unit  of  mass, 
at  the  same  distance,  will  then  be  expressed  by  the  mass  of  the 
sun  (M) ;  or,  in  other  words,  by  the  number  of  units  which  its 
mass  contains.  And  the  attraction  F,  at  any  distance  r,  will 

M. 
result  from  the  proportion  M  :  F::ra  :  I2,  which  gives  F=-^5 

This,  in  the  language  of  Dynamics,  is  the  Accelerating  Force  of 
the  planet,  due  to  the  attraction  of  the  sun. 

As  —  expresses  the  attraction  of  the  sun  for  a  unit  of  mass  of 
the  planet,  its  attraction  for  the  entire  mass  ra  of  the  planet  will 
be  expressed  by  ra  — .  This  'is  the  moving  force  of  the  planet, 

and  since  it  is,  at  the  same  distance,  proportional  to  the  mass  of 
the  planet,  the  velocity  due  to  its  action  is  the  same,  whatever 
may  be  the  mass. 

Attractive  Force  of  Planet.  The  planet  has  also  an  attraction 
for  the  sun,  as  well  as  the  sun  for  the  planet,  and  the  expression 
for  its  attractive  force,  or  for  the  accelerating  force  animating 

the  sun,  will  obviously  be  2?.     The  sun  will  then  tend  towards 

the  planet,  as  the  planet  towards  the  sun.  But  if  the  two  bodies 
were  to  set  out  from  a  state  of  rest,  the  velocity  of  the  planet 
would  be  as  many  times  greater  than  the  velocity  of  the  sun, 
as  the  mass  of  the  sun  is  greater  than  that  of  the  planet.  For 
the  velocity  of  the  planet  would  be  to  that  of  the  sun  as  the 
attractive  force  of  the  sun  is  to  the  attractive  force  of  the  planet, 

that  is,  as^:™,orasM:m. 

As  the  attractions  of  the  particles  of  the  sun  and  planet  are 
mutual  and  equal,  the  attraction  of  the  planet  for  the  entire 
mass  of  the  sun  must  be  equal  to  the  attraction  of  the  sun  for 
the  entire  mass  of  the  planet. 


GENERAL  PRINCIPLE  OF  REVOLUTION.  283 

491.  The  Sun  and  any  Planet  revolve  about  their 
Common  Centre  of  Oravity. 

To  show  this,  we  would  remark,  in  the  first  place,  that  it  is  a 
principle  of  Mechanics  that  the  mutual  actions  of  the  different 
members  of  a  system  of  bodies  cannot  affect  the  state  of  the 
centre  of  gravity  of  the  system.  This  is  called  the  Piinriple  of 
the  Preservation  of  the  Centre  of  Gravity.  It  follows  from  it  that 
the  common  centre  of  gravity  of  the  sun  and  any  planet  is  at 
rest,  unless  it  has  a  motion  of  translation  in  common  with  the 
two  bodies,  imparted  by  a  force  extraneous  to  the  system.  As 
we  are  concerned  at  present  only  with  the  relative  motion  of  the 
sun  and  planet,  such  motion  of  translation,  if  it  does  exist,  may 
be  left  out  of  account.  Now,  let  S  (Fig.  Ill) 
be  the  sun,  and  P  any  planet,  supposed  for 
the  moment  to  be  at  rest.  If  neither  of  the 
two  bodies  should  receive  a  velocity  in  a  di- 
rection inclined  to  PS,  the  line  of  their  cen- 
tres, they  would  move  towards  each  other  by 
virtue  of  their  mutual  attraction,  and  meet 
at  C  their  common  centre  of  gravity.*  But, 
if  the  body  P  have  a  projectile  velocity  given 
to  it  in  any  direction  P^,  inclined  to  the  line 
PS,  it  is  susceptible  of  proof  that  its  motion 
relative  to  the  sun  may  be  in  an  ellipse,  as  is 
observed  to  be  the  case  with  the  planets.  FIG.  in. 

Now,  while  the  planet  moves  in  space,  the 
line  of  the  centres  of  the  planet  and  sun  must  continually  pass 
through  the  stationary  position  of  the  centre  of  gravity ;  and 
therefore,  when  the  planet  has  advanced  to  any  point/),  the  sun 
will  have  shifted  its  position  to  some  point  s  on  the  line^C  pro- 
longed. Moreover,  as  the  two  bodies  mutually  gravitate  towards 
each  other,  the  path  of  each  in  space  will  be  continually  con- 
cave towards  the  other  body,  and  therefore  also  towards  the  cen- 
tre of  gravity  C,  which  is  constantly  in  the  same  direction  as  the 
other  body.  Since  the  planet  performs  a  revolution  around  the 
sun,  the  sun  and  planet  must  each  continue  to  move  about  the 
point  C  until  they  have  accomplished  a  revolution  and  returned 
to  the  line  PCS.  Also  as  the  distance  PS  of  the  two  bodies  will 
be  the  same  at  the  end  as  at  the  beginning  of  the  revolution,  as 
well  as  the  ratio  of  their  distances  PC  and  SC  from  the  centre 
of  gravity,  they  will  return  to  the  positions,  P,  S,  from 'which 
they  set  out,  and  will  therefore  move  in  continuous  curves. 

Moreover,  these  curves  are  similar  to  the  apparent  orbit  described  by  P  around 
S.  For,  draw  Sp'  parallel  and  equal  to  sp,  and  join  Pp  and  S*.  Then,  since 
sG  :  Cp ::  SC  :  CP,  Pp  is  parallel  to  Ss;  and  therefore  Pp  produced  passes  through 
p.  Whence,  CP  :  Cp::SP  :  Sp'.  Moreover,  the  angle  PCp  =  PS/.  It  follows, 

*  The  common  centre  of  gravity  of  two  bodies  lies  on  the  line  joining  their  cen- 
tres, and  divides  this  line  into  parts  inversely  proportional  to  the  masses  of  the 
bodies. 


284      THEORY  OF  THE   ELLIPTIC   MOTION  OF  THE  PLANETS. 

therefore,  that  the  area  PCp  is  similar  to  the  area  PSp' ;  and  thus  that  the  orbit 
of  P  around  C  is  similar  to  the  apparent  orbit  of  P  around  S.  The  latter  is  known 
from  observation  to  be  an  ellipse.  The  former  is  therefore  also  an  ellipse. 

As  the  distances  of  the  sun  and  planet  from  their  common 
centre  of  gravity  are  constantly  reciprocally  proportional  to 
their  masses,  the  orbit  of  the  sun  will  be  exceedingly  small  in 
comparison  with  the  orbit  of  the  planet. 

492.  Entire  Accelerating  Force  of  Planet.     If  to  both 
the  sun  and  planet  there  should  be  applied   a  force  equal  to 

the  accelerating  force  of  the  sun,  !^,  (49°),  but  in  an  opposite  di- 
rection, the  sun  would  be  solicited  by  two  forces  that  would 
destroy  each  other,  but  the  planet  would  now  be  urged  to- 
wards the  sun  remaining  stationary,  with  the  accelerating  force 

— i-?,  or  a  force  the  intensity  of  which  was  equal  to  the  sum 

of  the  intensities  of  the  attractive  forces  of  the  sun  and  planet, 
at  the  distance  of  the  planet.  Now,  the  application  of  a  common 
force  will  not  alter  the  relative  motion  of  the  two  bodies. 
Hence,  in  investigating  this  motion,  we  are  at  liberty  to  conceive 
the  sun  to  be  stationary,  if  we  suppose  the  planet  to  be  solicited 

by  the  accelerating  force  — it^.     As  the  mass  of  the  sun  is 

7*a 

very  much  greater  than  that  of  any  planet,  but  little  error  will 
be  committed  in  neglecting  the  attraction  of  the  planet,  and  tak- 
ing into  account  only  the  sun's  action  — . 

493.  General    Theoretical    Results.       Analysis    makes 
known  the  general  laws  of  the  motion  of  a  body,  when  im- 
pelled by  a  projectile  force,  and  afterwards  continually  attracted 
towards  the  sun's  centre  by  a  force  varying  inversely  as  the 
square  of  the  distance.     We  learn  by  it  that  the  body  will  neces- 
sarily describe  some  one  of  the  conic  sections  around  the  sun 
situated  at  one  of  its  foci.     We  learn,  also,  that  the  nature  of 
the  orbit,  as  well  as  the  length  of  the  major  axis,  is  wholly  de- 
pendent, for  any  given  distance  of  the  planet,  upon  the  intensity 
of  the  projectile  force ;  but  that  the  position  of  the  axis,  and  the 
eccentricity  of  the  orbit,  depend  also  upon  the  angle  of  projec- 
tion (that  is,  the  angle  included,  at  the  commencement  of  the 
motion,  between  the  line  of  direction  of  the  projectile  force  and 
the  radius-vector).      As  to  the  relative  intensity  of  projectile 
force  necessary  to  the  production  of  each  one  of  the  conic  sec- 
tions, a  certain  intensity  of  force  will  produce  a  parabola;  any 
less  intensity,  an  ellipse  or  circle ;  and  any  greater,  a  hvperbola. 

494.  Theoretical   Deter  initiation  of  Orbit  of  Planet. 
If  the  velocity  that  would  at  a  given  distance  be  imparted  by  the 
sun's  attraction  in  a  second  of  time,  which  is  the  measure  of  its 


r  *  ) 


THEORETICAL  DETERMINATION  OF  ORBIT.  285 

intensity  at  the  given  distance,  be  found,  and  also  the  distance  of 
a  planet  at  any  time,  as  well  as  its  velocity  and  the  angle  made 
by  the  direction  of  its  motion  with  the  radius-  vector,  the  form, 
dimensions,  and  position  of  the  planet's  orbit  can  be  computed. 
This  is  to  determine  the  orbit  a  priori.  The  practice  has  been, 
however,  to  determine  the  various  elements  of  a  planet's  orbit  by 
observation  (as  already  described,  Chap.  IX.). 

The  elements  being  known,  the  equations  of  the  elliptic  mo- 
tion, investigated  on  the  principles  of  Mechanics,  serve  to  make 
known  the  position  and  velocity  of  the  planet  at  any  time. 

The  physical  theory  of  the  motion  of  a  satellite  around  its 
primary  is  obviously  the  same  as  that  of  the  motion  of  a  planet 
around  the  sun. 

495.  Centre  of  Gravity  of  the  Solar  System.  According  to 
the  principle  of  the  preservation  of  the  centre  of  gravity  (491), 
the  centre  of  gravity  of  the  whole  solar  system  must  either  be 
at  rest,  or  have  a  motion  of  translation  in  space  in  common  with 
the  system,  resulting  from  the  action  of  a  foreign  force.     We 
have  already  seen  (447)  that  it  has  been  ascertained  from  observa 
tion,  that  it  is  in  fact  in  motion. 

The  sun  and  planets  revolve  around  their  common  centre  of 
gravity.  The  path  of  the  sun's  centre  results  from  the  joint  ac- 
tion of  all  the  planets,  and  is  a  complicated  curve.  As  the  quan- 
tity of  matter  in  all  the  planets  taken  together  is  very  small, 
compared  with  that  in  the  sun  (less  than  yj-g-),  the  extent  of 
the  curve  described  by  the  centre  of  the  sun  cannot  be  very- 
great.  It  is  found  by  computation,  that  the  distance  between 
the  sun's  centre  and  the  centre  of  gravity  of  the  system  can 
never  be  equal  to  the  sun's  diameter. 

496.  Centre  of  Gravity  of  a  Planet  and  its  Satellites*    It 
is  demonstrated  in  treatises  on  Mechanics,  that  if  foreign  forces 
act  upon  a  system  of  bodies,  the  centre  of  gravity  of  the  system 
will  move  just  as  the  whole  mass  of  the  system  concentrated  at 
the  centre  of  gravity  would  move,  under  the  action  of  the  same 
forces.     It  follows  from  this  principle,  that  from  the  attraction 
of  the  sun  for  a  primary  planet  and  its  satellites,  their  common 
centre  of  gravity  will  revolve  around  the  sun,  just  as  the  whole 
quantity  of  matter  in  the  planet  and  its  satellites  concentrated  at 
this   point  would,  under  the  influence  of  the  same  attraction. 
Moreover,  the  same  considerations  which  show  that  the  sun  and 
planets  revolve  about  their  common  centre  of  gravity,  will  also 
show  that  a  primary  planet  and  its  satellites  revolve  about  their 
common  centre  of  gravity.     It  appears,  therefore,  that  in  the 
case  of  a  planet  which  has  satellites,  it  is  not,  strictly  speaking,  the 
centre  of  the  planet  that  moves  agreeably  to  the  first  and  second 
laws  of  Kepler,  but  the  common  centre  of  gravity  of  the  planet 
and  its  satellites  ;  the  planet  and  satellites  revolving  around  the 
centre  of  gravity,  as  it  describes  its  orbit  about  the  sun. 


286      THEOKY  OF  THE  ELLIPTIC  MOTION  OF  THE  PLANETS. 

The  mass  of  the  earth  is  to  that  of  the  moon  as  82  to  1,  while 
the  distance  of  the  moon  is  to  the  radius  of  the  earth  as  60  to 
1 :  it  follows,  therefore,  that  the  common  centre  of  gravity  of  the 
earth  and  moon  lies  within  the  body  of  the  earth. 

497.  Kepler'§  third  Law  not  rigorously  true.  It  ap- 
pears from  the  physical  investigation  of  the  elliptic  motion  of 
the  planets,  that  Kepler's  third  law  is  not  strictly  true.  In  con- 
sequence of  the  action  of  the  planets  upon  the  sun,  the  ratio  of 
the  periodic  times  of  the  different  planets  depends  upon  the  mas- 
ses of  the  planets,  as  well  as  their  distances  from  the  sun.  If  p 
and  pr  be  the  periodic  times  of  any  two  of  the  planets,  a  and  a' 
their  mean  distances  from  the  sun's  centre,  and  m  and  m'  their 
quantities  of  matter,  that  of  the  sun  being  denoted  by  1,  then, 
disregarding  the  actions  of  the  other  planets,  theory  gives 

»•  :»*::_£!_ -_J!^ 

l+m     l+t»'- 

As  m  and  m'  are  very  small  fractions,  the  error  resulting  from 
their  omission  will  be  very  small.  If  we  omit  them,  we  shall 
have 

y  :/3::a':a"; 
which  is  Kepler's  third  law. 


PERTURBATIONS   OF  ELLIPTIC  MOTION  OF  THE  MOON.      287 


CHAPTER  XXIII. 

THEORY  OF  THE  PERTURBATIONS  OF  THE  ELLIPTIC  MOTION 
OF  THE  PLANETS  AND  THE  MOON. 


498.  WE  have,  in  a  previous  chapter,  given  a  general  idea  of  the  mode  of 
determining,  from  theory  and  observation  combined,  the  law  and  amount  of  the 
perturbations  or  inequalities  of  the  lunar  and  planetary  motions.     We  propose  now 
to  give  some  insight  into  the  nature  and  manner  of  operation  of  the  disturbing 
forces,  and  will  commence  with  the  perturbations  of  the  moon  produced  by  the 
action  of  the  sun. 

499.  Components  of  Disturbing  Force.    "We  have  already  shown  (209) 
how  the  intensity  and  direction  of  the  disturbing  force  of  the  sun,  in  any  given 
position  of  the  moon  in  its  orbit,  may  be  determined.     Let  us  now  derive  the 
disturbing  forces  that  take  effect  in  the  three  directions  in  which  the  motion 
of  the  moon  can  be  changed ;  namely,  in  the 

direction  of  the  radius-vector,  of  the  tangent 
to  the  orbit,  and  of  the  perpendicular  to  its 
plane.  Let  E  (Fig.  112)  be  the  earth,  M  the 
moon,  and  S  the  sun.  Let  the  force  exerted 
by  the  sun  upon  the  moon  be  decomposed  into 
two  forces,  one  acting  along  the  line  MS'  par- 
allel to  ES,  and  the  other  from  M  towards  E. 
If  the  component  along  MS'. were  equal  to  the 
force  exerted  by  the  sun  upon  the  earth,  the 
motion  of  the  moon  about  the  earth  would  not 
be  changed  by  the  action  of  these  two  forces. 
Hence,  the  difference  between  them  will  be  the 
disturbing  force  in  the  direction  MS'.  The 
component  along  ME  is  another  disturbing 
force.  It  is  called  the  Addititious  Force,  be- 
cause it  tends  to  increase  the  gravity  of  the 
moon  towards  the  earth.  The  disturbing  force 
along  MS'  will  generally  be  inclined  to  the 
plane  of  the  orbit,  and  may  be  decomposed 
into  three  forces,  one  in  the  direction  of  the 
tangent,  another  in  the  direction  of  the  radius- 
vector,  and  a  third  in  the  direction  of  the  per- 
pendicular to  the  plane.  The  first  mentioned 
component  is  called  the  Tangential  Force;  the 
second  is  called  the  Ablatitious  Force ;  and  the 
third  we  shall  call  the  Perpendicular  Force. 

The  actual  disturbing  force  in  the  direction  of  the  radius-vector  is  equal  to  the 
difference  between  the  addititious  and  ablatitious  forces,  and  is  called  the  Radial 
Force.  This  and  the  tangential  and  perpendicular  forces  constitute  the  disturbing 
forces,  the  direct  operation  of  which  is  to  be  considered. 

500.  To  obtain  General  Analytical  Expressions  for  these  Forces, 
let  the  distance  of  the  sun  from  the  earth  (which  for  the  present  we  shall  suppose 
to  be  constant)  be  denoted  by  a,  and  the  distances  of  the  moon  from  the  earth  and 
sun,  respectively,  by  y  and  z.    Also  let  F  =  the  force  exerted  by  the  earth  upon 
the  moon,  P  =  the  force  exerted'  by  the  sun  upon  the  earth,  and  Q  =  the  force 


FlG.  112. 


288      PERTURBATIONS  OF  ELLIPTIC   MOTION   OF  THE  MOON. 

exerted  by  the  sun  upon  the  moon.    Then,  if  we  denote  the  mass  of  the  earth  by 
1,  and  take  m  to  stand  for  the  mass  of  the  sun,  we  shall  have  (490), 


Let  the  force  Q  be  represented  by  the  line  MS  (Fig.  112);  and  let  its  component 
parallel  to  BS,  or  MS'—  R,  and  its  component  along  the  radius-  vector,  or  ME=T. 


Q  :  T  :  :  MS  :  ME;  or,  ™  :  T  :  :  z  : 


Whence, 


addititious  force  T  =  IIT*. . .  .(82). 

z3 
In  a  similar  manner  we  obtain 

E_  ma        ,Qa» 
_  _ (83). 

The  disturbing  force  in  the  direction  of  the  sun 

"^""a9^       \?~~a»)' 

Now,  let  a,  #,  y,  denote  the  angles  made  by  the  line  MS'  with  the  tangent,  radius- 
vector,  and  perpendicular  to  the  plane  of  the  orbit,  and  we  shall  have  for  the  ap- 
proximate components  of  the  disturbing  force  R  —  P,  along  these  lines : 

tangential  force  =  ma  (  _  —  _  j  cos  o . . . .  (84) ; 

ablatitious  force  =  ma  (  1  —  L  \  cos  /? (85); 

\  zs      a* ) 

perpendicular  force  =  ma  (  —  —  -  ^  cos  y. ... (86). 

\  zs      a8  ) 

Combining  equ.  (85)  with  equ.  (82),  we  obtain  for 
the  radial  force, 

radial  force =my—  —  ma  (  —  —  _  ]  cos  /?. 
z1  V  z3      a8 ) 

The  obliquity  of  the  orbit  of  the  moon  to  the 
plane  of  the  ecliptic,  affects  but  very  slightly  the 
value  of  the  tangential  and  radial  forces.  If  we 
leave  it  out  of  account,  or  suppose  the  moon's 
orbit  to  lie  in  the  plane  of  the  ecliptic,  we  shall 
have  (Fig.  113)  ,8  —  S'ML  =  SEM,  the  elongation 
of  the  moon  =  <p,  and  a  =  complement  of  </>,  which 
gives 

tang,  force  =  ma  (  _  —  _  J  sin  A (87) ; 

rad.  force  =  my  _  —  ma  (  _  —  _  )  cos  A  (88). 
z8  \  z3      o?  ) 

Equation  (86)  may  be  transformed  into  another 
which  is  better  adapted  to  the  purposes  we  have 
in  view.  Let  MK  (Fig.  112)  represent  the  per- 
pendicular to  the  plane  of  the  moon's  orbit,  MF 
the  intersection  of  the  plane  SMK  with  the  plane 
FIQ.  113.  of  the  moon's  orbit,  and  SI,  IF,  the  intersections 

of  a  plane  passing  through  S  and  perpendicular 

to  EN,  the  line  of  nodes,  with  the  plane  of  the  ecliptic  and  the  plane  of  the  orbit. 
SF  will  be  perpendicular  to  both  IF  and  MF.     Denote  SIF,  the  inclination  of  the 
orbit  to  the  ecliptic,  by  I,  SEN  the  angular  distance  of  the  sun  from  the  node  by 
N,  and  SE  and  SM  by  a  and  z,  as  before. 
Now,  in  equ.  (86)  y  stands  for  the  angle  S'MK,  but  S'MK  =  SMK  (nearlyX  and 

cos  SMK  =  sin  SMF  =  —. 
SM 


INVESTIGATION  OF  THE   DISTURBING   FORCES.  289 

SF  =  SI  sin  SIP,  and  SI  =  SE  sin  SEI  ; 

whence  SF  =  SE  sin  SEI  sin  SIF  =  a  sin  N  sin  I  : 

substituting, 

cos  y  =  cos  SMK  =  as 

SM 

Thus  we  have 


force  =  ma  (  1  -  1  ^  ^!?J*_!™!.  .  .  .(89). 
\  3s      a  *J  z 


perpen. 

The  variable  z  may  be  eliminated  from  equations  (87),  (88),  and  (89),  and  ocher 
equations  obtained,  involving  only  the  variables  y  and  0.  Let  ML  (Fig.  112)  be 
drawn  through  the  place  of  the  moon  perpendicular  to  ES.  Then,  using  the  same 
notation  as  before, 

LS  =  2  (nearly),  EL  =  EM  cos  LEM  =  y  cos  $. 
But  LS=SE  —  EL; 

vrhence  z  =  a  —  y  cos  <j>,  and  z3  =  a3  —  3a2y  cos  ^  : 

neglecting  the  terras  containing  the  higher  powers  of  y  than  the  first,  as  they  are 
very  minute,  y  being  only  about  5-^  a. 

Thus,  1=  _  L_     _  -  1  +  3y  cos  ft  . 

2*      a3  —  3a2?/  cos  <£      a3          a4 
neglecting  all  the  terms  of  the  quotient  that  involve  higher  powers  of  y  than  the 

first.    Substituting  this  value  of_  in  equ.  (87),  we  obtain, 

2s 

tangential  force  = 
or  (App.  For.  13), 


tangential  force  =  -  .  .  .(90). 

Making  the  same  substitution  in  equ.  (88),  and  neglecting  the  term  containing  y*, 
there  results, 

radial  force  =  ™V  (l-3cogV). 

or  (App.  For.  9), 

radial  force  =  -  m*  <*  +  l^L2*?  .  .  .  .(91). 
2  a9 

In  equ.  (89)  we  have  to  substitute,  besides,  the  value  of  z,  viz.  a  —  y  cos  <f>  ;  then 
dividing  and  neglecting  as  before,  we  have 

perpen.  force  =  ^V^±  sin  tf  sin  I.  .  .  .(92). 
a* 

5  0  1  .  Variations  of  disturbing  forces.  If  the  disturbing  forces  retained  constantly 
the  same  intensity  and  direction,  the  result  would  be  a  continual  progressive  de- 
parture from  the  elliptic  place  ;  but,  in  point  of  fact,  these  forces  are  subject  to 
periodical  changes  of  intensity  and  direction  from  several  causes,  from  which  re- 
sults a  compensation  of  effects,  and  an  eventual  return  to  the  elliptic  place.  The 
causes  of  the  variation  of  the  disturbing  forces  are  : 

(1.)  The  revolution  of  the  moon  around  the  earth. 

(2.)  The  elliptic  form  of  the  apparent  orbit  of  the  sun. 

(3.)  The  elliptic  form  of  the  orbit  of  the  moon. 

(4.)  The  inclination  of  the  two  orbits. 

As  the  variations  of  the  radial  and  tangential  forces,  resulting  from  the  inclina- 
tion of  the  orbits,  are  very  minute,  we  shall  leave  them  out  of  account,  and  hi  the 
consideration  of  the  effects  of  these  forces  shall,  for  the  sake  of  simplicity,  regard 
the  orbits  as  lying  hi  the  same  plane. 

The  first  mentioned  circumstance  is  the  most  prominent  cause  of  variation,  and 
gives  rise  to  the  more  conspicuous  perturbations.  The  other  two  serve  to  modify 

19 


290      PERTUKBATIONS   OF   ELLIPTIC   MOTION   OF  THE   MOON. 


the  variations  of  the  forces  resulting  from  the  first,  and  occasion  each  a  distinct  set 
of  periodical  perturbations. 

5O2.  Tangential  Force.  Let  us  now  investigate,  in  succession,  the  effects 
of  each  of  the  disturbing  forces,  commencing  with  the  tangential  force.  The  tan- 
gential force  takes  effect  directly  upon  the  velocity  of  the  moon  in  its  orbit ;  and 
as  its  line  of  direction  does  not  pass  through  the  earth,  it  disturbs  the  equable  des- 
cription of  areas.  It  also  affects  the  radius-vector  indirectly,  by  changing  the  cen- 
trifugal force.  To  understand  the  detail  of  its  action  we  must  inquire  into  the 
variations  which  it  undergoes. 

If  we  regard  y  as  constant  in  the  expression  for  the  tangential  force,  (equa.  90), 
which  amounts  to  considering  the  moon's  orbit  as  circular,  the  expression  will  be- 
come equal  to  zero  when  sin  2/-=0,  and  will  have  its  maximum  value  when  sin 
2^=1.  It  will  also  change  its  sign  with  sin  2p.  It  appears,  therefore,  that  the 
tangential  force  is  zero  in  the  syzigies  and  quadratures,, where  it  also  changes  its 
direction,  and  that  it  attains  its  maximum  value  in  the  octants.  It  will  be  seen,  on 

inspecting  Fig.  1 14,  that  it  will  be  a  retard- 
ing force  in  the  first  quadrant  (AB).  Ac- 
cordingly, it  will  be  an  accelerating  force  in 
the  second,  a  retarding  force  again  in  the 
third,  and  an  accelerating  force  again  in  the 
fourth. 

This  will  also  appear  upon  considering 
the  direction  of  the  disturbing  force  parallel 
to  the  line  of  the  centres  of  the  sun  and 
earth,  in  the  various  quadrants.  In  the 
nearer  half  of  the  orbit  the  sun  tends  to 
draw  the  moon  away  from  the  earth,  and  the 
force  in  question  is  directed  towards  the  sun. 
In  the  more  remote  half  the  sun  tends  to 
draw  the  earth  away  from  the  moon,  but  we 
may  regard  it,  instead,  as  urging  the  moon 
from  the  earth  by  the  same  force ;  for  the 
relative  motion  will  be  the  same  on  this  supposition.  In  the  part  of  the  orbit  sup- 
posed, then,  the  disturbing  force  under  consideration  will  be  directed  from  the  sun, 
as  represented  in  Fig.  114. 

It  appears,  then,  that  the  tangential  force  will  alternately  retard  and  accelerate 
the  motion  of  the  moon  during  its  passage  through  the  different  quadrants,  and 
that  the  maximum  of  velocity  will  occur  in  the  syzigies,  A,  C,  where  the  accelerat- 
ing force  becomes  zero,  and  the  minimum  of  velocity  in  the  quadratures,  B,  D,  where 
the  retarding  force  becomes  zero.  On  the  supposition  that  the  orbit  is  a  circle,  the 
arcs  AB,  BO,  CD,  and  DA,  would  be  equal,  and  the  retardation  of  the  velocity  in 
one  quadrant  would  be  compensated  for  by  an  equal  acceleration  in  the  next,  and 
at  the  close  of  a  synodic  revolution  the  velocity  of  the  moon  would  be  the  same  as 
at  its  commencement.  As  the  velocity  is  greatest  in  the  syzigies  and  least  in  the 
quadratures,  and  as  the  degree  of  retardation  is  the  same  as  that  of  acceleration, 
the  mean  motion*  must  have  place  in  the  octants.  Now,  as  the  moon  moves  from 
the  syzigy  A  with  a  motion  greater  than  the  mean  motion,  its  true  place  will  be 
in  advance  of  its  mean  place,  and  will  become  more  and  more  so  till  it  reaches 
the  octant,  where  the  true  motion  is  equal  to  the  mean.  The  difference  between 
the  true  and  mean  place  will  then  be  the  greatest ;  for  after  that,  the  true  motion 
becoming  less  than  the  mean,  the  mean  place  will  approach  nearer  to  the  true,  till 
at  the  quadrature  they  coincide.  Beyond  B,  the  true  motion  still  continuing  less 
than  the  mean,  Ihe  mean  place  will  be  in  advance  of  the  true,  and  the  separation 
will  increase  till  at  the  octant  the  true  motion  has  attained  to  an  equality  with 
the  mean  motion,  after  which,  the  mean  motion  being  the  slowest,  the  true  place 
will  approach  the  mean  till  at  the  syzigy  C  they  again  coincide.  Corresponding 
effects  will  take  place  in  the  two  remaining  quadrants.  We  perceive,  therefore, 
that  the  tangential  force  produces  an  inequality  of  longitude,  which  attains  to  its 
maximum  positive  and  negative  value  in  the  octants,  and  is  zero  in  the  syzigies. 


FIG.  114. 


*  The  expressions,  mean  motion,  true  motion,  mean  place,  true  place,  are  here  to 
be  understood  only  in  relation  to  the  perturbation  under  consideration. 


EFFECTS  OF  THE  TANGENTIAL  FORCE.        291 

This   is  the  inequality  known  in  Spherical  Astronomy  by  the  name  of  Varia* 
tow  (2 1 7). 

503.  Modifications  of  the  effects  of  the  tangential  force,  that  result  from  the  elliptic 
form  of  the  sun's  orbit.    Suppose  that  at  the  moment  when  the  moon  sets  out  from 
conjunction  the  sun  is  in  the  apogee  of  its  orbit :  then  it  is  plain  that,  during  the 
whole  revolution  of  the  moon,  the  sun's  disturbing  force  would  be  on  the  increase  by 
reason  of  the  diminution  of  the  sun's  distance,  and  that,  in  consequence,  the  retaru- 
ation  in  the  first  quadrant  would  be  less  than  the  acceleration  in  the  second,  and 
the  retardation  in  the  third  less  than  the  acceleration  in  the  fourth.     So  that,  when 
the  moon  has  again  come  round  into  conjunction,  the  acceleration  will  have  over- 
compensated  the  retardation.      This  kind  of  action  would  go  on  so  long  as  the  sun 
approaches  the  earth;  but  when  it  has  passed  the  perigee  of  its  orbit,  and  begun  to 
recede  from  the  earth,  the  reverse  effect  would  take  place,  and  a  retardation  of  the 
moon's  orbital  motion  would  happen  each  revolution.     If  the  anomalistic  revolution 
of  the  sun  were  an  exact  multiple  of  the  synodic  revolution  of  the  moon,  the  accelera- 
tion in  each  revolution  of  the  moon  during  the  passage  of  the  sun  from  the  apogee 
to  the  perigee  of  its  orbit,  would  be  compensated  for  by  an  equivalent  retardation 
in  the  revolution  of  the  moon,  answering  to  the  same  distance  of  the  sun  in  its 
passage  from  the  perigee  to  the  apogee;    and  the  velocity  of  the  moon  would  be  the 
same  at  the  close  of  an  anomalistic  revolution  of  the  sun  as  at  its  commencement. 
But  as  this  relation  does  not,  in  fact,  subsist  between  the  anomalistic  revolution  of 
the  sun  and  the  synodic  revolution  of  the  moon,  a  compensation  between  the  accele- 
rations and  retardations,  answering  to  the  different  revolutions  of  the  moon,  will 
not  be  effected  until  conjunctions  shall  have  occurred  at  every  variety  of  distance 
of  the  sun  in  each  half  of  its  orbit     Since  the  anomalistic  and  synodic  revolutions 
are  incommensurable,  the  sun  will  be,  in  reality,  in  every  variety  of  position  in  its 
orbit  at  the  time  of  conjunction,  in  process  of  time,  so  that  eventually  the  original 
velocity  in  conjunction  will  be  regained.     It  appears,  therefore,  that  the  variation  of 
the  moon's  motion  from  one  revolution  to  another,  occasioned  by  the  elliptic  form 
of  the  sun's  orbit,  is  periodic.     Its  period  will  be  the  interval  of  time  in  which  the 
moon  will  perform  a  certain  number  of  synodic  revolutions,  while  the  sun  performs 
a  certain  number  of  anomalistic  revolutions.     Avoiding  unnecessary  precision,  we 
find  it  to  consist  of  but  a  moderate  number  of  years. 

504.  Consequences  of  the  elliptic  form  of  the  moon's  orbit.    "We  remark,  in  the  first 
place,  that  the  orbit  being  an  ellipse,  the  areas  AEB,  BEG,  CED,  and  DBA  (Fig. 
114).  will  be  unequal,  and  therefore,  by  the  laws  of  elliptic  motion,  the  arcs  AB,  BC, 
CD,  and  DA,  will  be  described  in  unequal  times.    It  follows  from  this,  that  the  retar- 
dation in  the  first  quadrant  will  not  be  exactly  compensated  by  the  acceleration  in 
the  second,  and  that  the  retardation  in  the  third  will  not  be  exactly  compensated  by 
the  acceleration  in  the  fourth.     Therefore,  at  the  end  of  the  synodic  revolution  the 
moon  will  have  an  excess  or  deficiency  of  velocity.     Its  mean  motion  will  then  vary 
from  one  revolution  to  another,  by  reason  of  the  ellipticity  of  its  orbit.     This  varia- 
tion will  be  periodic/like  that  just  considered,  and  for  similar  reasons.     The  excess 
or  deficiency  of  velocity  at  the  close  of  any  one  revolution,  will  in  time  be  compen- 
sated by  an  equal  deficiency  or  excess  occurring  at  the  close  of  another  revolution, 
when  the  sun  has  a  certain  different  position  with  respect  to  the  perigee  of  the  moon's 
orbit. 

505.  Radial  Force.     "We  pass  now  to  the  consideration  of  the  action  of  the 
radial  force.      The  direct  general  effect  of  the  radial  force,  is  an  alteration  in  the 
intensity  of  the  moon's  gravity  towards  the  earth,  and  in  its  law  of  variation.     Its 
specific  effects   are  periodical  variations  in  the  magnitude,  eccentricity,  and  position 
of  the  orbit.     As  it  is  directed  towards  the  earth,  it  will  not  disturb  the  equable 
description  of  areas.     To  discover  the  variations  of  this  force  we  have  only  to  dis- 
cuss the  general  analytical  expression  for  it,  already  investigated.     It  is, 

radial  force  = 

"We  shall  have  radial  force  =  0,  when  1  —  3  cos*  0  =  0,  or  when  cos  <p  = 
This  value  of  cos  <p  answers  to  four  points  lying  on  either  side  of  the  quadratures, 
and  about  35°  distant  from  them.  When  cos  <j>  is  numerically  greater  than  ,/£  the 
result  will  be  negative,  and  when  it  is  less  than  ^/^  the  result  will  be  positive.  It 
follows,  therefore,  that  the  radial  force  increases  the  gravity  of  the  moon  in  th« 


292      PERTURBATIONS   OF   ELLIPTIC   MOTION   OF  THE   MOON. 

quadratures,  and  for  about  35°  on  each  side  of  them,  and  that  during  the  remainder 
of  a  synodic  revolution  it  diminishes  it. 

When  the  moon  is  in  quadratures,  cos  <£  =  0,  and 

radial  force  =-^L (93). 

a3 

In  the  syzigies,  we  have  cos  <p  =  ±  1,  which  gives 

radial  force  -—^. .  .  .  (94). 
a" 

It  appears,  then,  that  the  diminution  of  the  moon's  gravity  in  the  syzigies  is  dou- 
ble of  iis  increase  in  the  quadratures. 

We  learn  also  from  equations  (93j  and  (94),  that  the  radial  force  in  the  quadra- 
tures and  syzigies  varies  directly  as  the  distance;  from  which  we  conclude  that  the 
gravity  of  the  moon  varies  at  these  points  by  a  different  law  from  that  of  the  inverse 
squares  In  the  quadratures  the  gravity  will  be  increased  most  at  the  greatest 
distance,  where  it  is  the  least ;  and  thus  it  will  vary  in  a  less  rapid  ratio  than  the 
square  of  the  distance.  In  the  syzigies  it  will  be  diminished  most  at  the  greatest 
distance,  or  where  it  is  the  least;  and  accordingly,  at  these  points  it  will  vary  in  a 
more  rapid  ratio  than  the  square  of  the  distance. 

506.  Maoris  distance'* increased  by  radial  force.      With  the  aid  of  the  Differ- 
ential Calculus,  we  readily  rind  that  the  mean  diminution  of  the  moon's  gravity 

from  the  sun's  action  is  ^ ;  r  representing  in  this  case  the  mean  distance  of  the 

2a3 

moon  from  the  earth.  The  value  of  this  expression  is  equal  to  about  the  360th 
part  of  the  whole  gravity  of  the  moon  to  the  earth. 

In  consequence  of  this  diminution,  the  moon  must  describe  its  orbit  at  a  greater 
distance  from  the  earth,  with  a  less  angular  velocity,  and  in  a  longer  time,  than  if 
it  were  acted  on  only  by  the  attraction  of  the  earth. 

507.  The  radial  force  of  the  sun  alters  the  eccentricity  of  the  moon's  orbit  and 
differently  in  different  revolutions  of  the  moon,  according  to  the  position  of  the  line 
of  syzigies  with  respect  to  the  line  of  apsides.     When  these  lines  are  coincident  the 

eccentricit}'  is  increased.  For  suppose  PMAN 
(Fig.  115)  to  be  the  elliptic  orbit  of  the  moon 
that  would  be  described  under  the  influence 
of  a  force  varying  inversely  as  the  square  of 
the  distance.  In  going  from  the  apogee  to 
the  perigee,  the  gravity  will  increase  in  a 
greater  ratio  than  that  of  the  inverse  square 
of  the  distance ;  the  true  orbit  will  therefore 
fall  within  the  ellipse,  and  the  perigean  dis- 
tance (EP')  will  be  Jess  than  for  the  ellipse. 
Consequently,  the  eccentricity  will  increase 
so  much  the  more  as  the  major  axis  dimin- 
ishes. On  the  other  hand,  in  going  from  the 
FIG.  115.  perigee  to  the  apogee,  the  gravity  will  de- 

crease in  a  greater  ratio  than  the  inverse 

square  of  the  distance,  and  the  moon  will  consequently  recede  further  from  the 
earth  than  if  the  orbit  described  was  an  ellipse.  Therefore,  in  this  half  of  the  orbit 
the  eccentricity  will  also  be  increased.  When  the  apsides  are  in  quadratures 
the  eccentricity  will  be  diminished ;  for  the  gravity  will  then  vary  from  the  apogee 
to  the  perigee,  and  from  the  perigee  to  the  apogee,  in  a  less  ratio  than  that  of  the 
inverse  squares ;  and  therefore  the  results  will  be  contrary  to  those  just  obtained. 
The  eccentricity  will  have  its  maximum  value  when  the  apsides  are  in  syzigies,  and 
its  minimum  when  they  are  in  quadratures  ;  for,  in  every  other  position  of  the  line  of 
apsides  with  respect  to  the  line  of  syzigies,  the  radial  force  in  the  apogee  and  peri- 
gee will  be  less  than  in  these  positions  (equa.  91),  and  therefore  alter  less  the  pro- 
portional gravity  of  the  moon  in  the  apogee  and  perigee.  It  is  evident,  from  the 
gradual  decrease  of  the  radial  force  as  we  recede  from  the  syzigies  and  quadratures, 
that  the  eccentricity  will  continually  diminish  in  the  progress  of  the  apsides  from 
the  syzigies  to  the  quadratures,  and  that  it  will  continually  increase  from  the  quadra- 
tures to  the  syzigies. 


EFFECTS  OF  THE   RADIAL  FORCE.  293 

The  change  in  the  eccentricity  of  the  moon's  orbit,  thus  produced,  will  be 
ottended  with  a  corresponding  change  in  the  equation  of  the  centre,  and  thus  of  the 
longitude  And  this  change  is  the  conspicuous  inequality  of  the  moon,  known  by 
the  name  of  Kvection  (217). 

508.  The  radud  force  also  produces  a  motion  of  the  line  of  apsides.     If  the  moon 
were  only  acted  upon  by  the  attraction  of  the  earth  its  orbit  would  be  an  ellipse, 
and  the  moduli  from  one  apsis  to  another,  or,  in  other  words,  from  one  point  where 
the  orbit  cuts  the  radius-vector  at  right  angles  to  the  other,  would  be  180°.     In 
point  of  fact,  however,  the  gravity  due  to  the  earth's  attraction  is  constantly  either 
diminished  or  increased  by  the  radial  disturbing  force  of  the  sun,  and  therefore  its 
true  orbit  must  continually  deviate  from  the  ellipse  that  would  be  described  under 
the  sole  action  of  the  earth's  attraction.     When  from  the  action  of  this  force  there 
is  a  diminution   of  the  force  of  gravity,  the  moon  will  continually  recede  from 
the  ellipse  in  question,  its  path  will  be  less  bent,  and  it  must  therefore  move  through 
a  greater  angular  distance  before  the  central  force  will  have  deflected  its  course  into 
a  direction  at  right  angles  to  the  radius-vector.     Accordingly,  it  will  move  through 
a  greater  angular  distance  than  180°  in  going  from  one  apsis  to  another,  and  thus 
the  apsides  will  advance.     On  the  other  hand,  when  the  same  force  increases  the 
force  of  gravity,  the  moon's  path  will  fall  within  the  ellipse,  its  curvature  will  be 
increased,  and  therefore  it  will  be  brought  to  intersect  the  radius-vector  at  right 
angles  at  a  less  angular  distance.     In  this  case,  therefore,  tho  apsides  will  move 
backward.     Now,  we  have  shown  (505)  that  the  radial  disturbing  force  of  the  sun 
alternately  diminishes  and  increases  the  moon's  gravity  to  the  earth.     It  follows, 
therefore,  that  the  motion  of  the  apsides  will  be  alternately  direct  and  retrograde; 
but  since,  as  has  been  shown  (505),  the  diminution  subsists  during  a  longer  part  of 
the  moon's  revolution,  and  is  moreover  greater  than  the  increase,  the  direct  motion 
will  exceed  the  retrograde,  and  therefore  in  an  entire  revolution  the  apsides  will 
advance. 

The  observed  motion  of  the  apsides  of  the  moon's  orbit  is  not,  however,  wholly 
produced  by  the  radial  disturbing  force.  It  is  in  part  due  to  the  action  of  the  tan- 
gential force.  This  force  alters  the  centrifugal  force  of  the  moon,  and  thus  changes 
its  gravity  towards  the  earth,  at  the  same  time  with  the  radial  force. 

509.  Explanation  of  the  Annual  Equation.     The  elliptic  form  of  the  sun's  orbit 
is  the  occasion  of  a  change  in  the  radial  force,  from  which  results  a  perturbation  of 
longitude  called  the  Annual  Equation  (217).     The  mean  diminution  of  the  moon's 
gravity,  arising  from  the  action  of  the  sun,  or  the  mean  radial  force,  is  equal  to 

!^  (506).     Hence  this  diminution  is  inversely  proportional  to  the  cube  of  the  sun's 

distance  from  the  earth.  Therefore,  as  the  sun  approaches  the  perigee  of  its  orbit, 
its  distance  from  the  earth  diminishing,  the  mean  diminution  of  the  moon's  gravity 
to  the  earth  will  increase,  arid  consequently  the  moon's  distance  from  the  earth  will 
become  greater,  and  its  motion  slower,  than  it  otherwise  would  be.  The  contrary 
will  take  pUce  while  the  sun  is  moving  from  the  perigee  to  the  apogee. 

510.  Perpendicular  Force.     The  disturbing  force  perpendicular  to  the  plane 
of  the  moon's  orbit,  produces  a  tendency  in  the  moon  to  quit  that  plane,  from  which 
there  results  a  change  in  the  position  of  the  line  of  the  nodes,  and  a  change  in  the 
inclination  of  the  plane  of  the  orbit  to  that  of  the  ecliptic.     If  we  examine  the 
general  expression  tor  this  force,  viz.: 


perpen.  force=3my  f^an  N  sin  I, 


we  see  that  for  any  given  values  of  N  and  I,  it  will  be  zero  in  the  quadratures,  and 
have  its  greatest  value  in  the  syzigies;  and  that  it  will  change  its  direction  in  the 
quadratures,  lying,  in  the  nearer  half  of  the  orbit,  on  the  same  side  of  its  plane  as 
the  sun.  and  in  the  more  remote  half,  on  the  opposite  side.  We  perceive  also  that 
k  will  be  zero  for  every  value  of  0,  or  for  every  elongation  of  the  moon,  when  the 
angle  N  is  zero,  tbat  is,  when  the  sun  is  in  the  plane  of  the  orbit;  and  will  attain 
its  maximum,  for  any  given  elongation,  when  the  line  of  direction  of  the  sun  is  per- 
pendicular to  the  line  of  nodes.  It  will  also  be  the  less,  other  things  being  the 
game,  the  smaller  is  the  inclination  I. 

511.     Retrograde  Motion  of  the  Nodes.     Let  NM'B  (Fig.  116)  represent  the  orbit 


291      PERTURBATIONS  OF  ELLIPTIC   MOTION   OF  THE   MOON. 


FIG.  116. 


of  the  moon,  and  S  the  sun,  supposed  stationary,  the  line  of  the  nodes  being  in  quadra- 
tures ;  and  let  L,  L'  be  the  points  of  the  orbit  90° 
distant  from  the  nodes.  The  direction  of  the 
force,  in  the  various  points  of  the  orbit,  is  in- 
dicated  by  the  arrows  drawn  in  the  figure. 
When  the  moon  is  at  any  point  M'  between  L 
and  the  descending  node  N',  it  will  be  drawn 
out  of  the  plane  in  which  it  is  moving  by  the 
disturbing  force  M'K',  and  compelled  to  move 
in  such  a  line  as  MY.  The  node  N'  will  there- 
fore retrograde  to  some  point  ri.  When  the 
moon  is  at  any  point  M,  moving  from  the  as- 
cending node  N  towards  L,  its  course  will  be 
changed  to  the  line  M/,  lying,  like  the  line  MY, 
below  the  orbit,  which  being  produced  back- 
ward, meets  the  plane  of  the  ecliptic  in  some 
point  n  behind  N.  The  nodes,  therefore,  retro- 
grade in  this  position  of  the  moon,  as  well  as  in 
the  former.  When  the  moon  is  in  the  half 
N'L'N  of  the  orbit,  lying  below  the  ecliptic, 
the  absolute  direction  of  the  disturbing  force 

will  be  reversed,  and  thus  its  tendency  will  be  the  same  as  before,  namely,  to 
draw  the  moon  towards  the  ecliptic.  It  follows,  therefore,  that  throughout  this 
half  of  the  orbit,  as  in  the  other,  the  motion  of  the  nodes  will  be  retrograde.  Ac- 
cordingly, when  the  nodes  are  in  quadratures,  or  90°  distant  from  the  sun,  they  will 
retrograde  during  every  part  of  the  revolution  of  the  moon. 

Suppose  the  sun  now  to  be  fixed  on  the  line  of  nodes,  or  the  nodes  to  be  in 
syzigies.  In  this  case  the  perpendicular  force  will  be  zero  (510),  and  therefore 
there  will  be  no  disturbance  of  the  plane  of  the  moon's  orbit. 

Next,  let  the  situation  of  the  sun  be  intermediate  between  the  two  just  consid- 
ered, as  represented  in  Figs.  116  and  117.  The  effect  of  the  disturbing  force  will 
be  the  same  as  in  the  first  situation  from  the  quadrature  q  (Fig.  116)  to  the  node 
N",  and  from  the  quadrature  q  to  the  node  N.  But  throughout  the  arcs  Ng,  NY, 
the  direction  of  the  force,  and  therefore  the  effects,  will  be  reversed.  The  node 
will  then  retrograde,  as  before,  while  the  moon  moves  over  the  arcs  qN'  and  q  'N, 
and  advance  while  it  is  in  the  arcs  N#,  N'g'.  But  as  the  force  is  greatest  over  the 
arcs  <?N',  <?'N,  which  contain  the  syzigies  (510),  and  as  these  arcs  are  also  longer 
than  the  arcs  N#,  N'q',  the  node  will,  on  the  whole,  retrograde  each  revolution. 
The  velocity  of  retrogradation  will,  however,  be  less  than  when  the  nodes  are  in 
quadratures,  and  proportionably  less  as  the  distance  of  the  sun  from  this  position 
is  greater. 

In  the  position  represented  in  Fig.  117,  a 
direct  motion  will  take  place  over  the  area 
q'N'  and  qN :  but  as  Ng'  and  N'<?,  the  area 
of  retrograde  motion,  are  of  greater  extent 
than  <?'N'  and  <?N,  and  moreover  contain  the 
syzigies,  the  retrograde  motion  in  each  revo- 
lution must  exceed  the  direct,  as  before. 

If  we  suppose  the  sun  to  be  situated  on 
the  other  side  of  the  line  of  nodes,  the  effect 
of  the  disturbing  force  will  obviously  be  the 
same  in  any  one  position  of  the  sun,  as  in  the 
position  diametrically  opposite  to  it.  It  ap- 
pears, then,  that  the  line  of  the  nodes  has  a 
retrograde  motion  in  every  possible  position 
of  the  sun. 

512.  Effect  of  5«w's  Motion.  We  have 
thus  far  supposed  the  sun  to  remainstation- 
ary  in  the  various  positions  in  which  we 
have  considered  it,  during  the  revolution  of 
the  moon.  It  remains,  then,  to  consider  the 
effect  of  the  sun's  motion  in  this  interval 
And  first,  it  is  plain,  that,  as  the  sun  advances  from  S  towards  N"  (Fig.  116),  the 


FIG.  117. 


EFFECTS  OF  THE  PERPENDICULAR  FORCE. 


295 


arcs  Ng,  N'q'  will  increase,  and  the  arcs  qN'  and  g'N  diminish ;  from  which  it  ap- 
pears, that  during  the  advance  of  the  sun  from  the  point  90°  behind  the  descending 
node  to  this  node,  its  motion  in  the  course  of  each  revolution  of  the  moon  will 
cause  the  retrograde  motion  of  the  node  to  be  slower  than  it  otherwise  would  be. 
While  the  sun  moves  from  the  ascending  node  to  the  point  90°  from  it,  the  effect  of 
its  motion  will  obviously  be  just  the  reverse  of  this.  During  its  passage  from  the 
descending  to  the  ascending  node,  the  effect  will  be  the  same  in  either  quadrant 
as  in  that  diametrically  opposite. 

The  variation  in  the  intensity  of  the  perpendicular  force,  conspires  with  the  dif- 
ference of  situation  of  the  sun  and  its  motion  during  a  revolution  of  the  moon  in 
diminishing  or  increasing,  as  the  case  may  be,  the  velocity  of  retrogradation  of 
the  nodes. 

513.  Change  of  the  inclination  of  the  orbit.  If  we  refer  to  Fig.  116  we  shall  see 
that  when  the  nodes  are  in  quadrature  the  inclination  will  diminish  while  the  moon 
is  moving  from  the  ascending  node  N  to  the  point  L  90°  distant  from  it,  and  in- 
crease while  it  is  moving  from  L  to  the  other  node  N".  In  the  other  half  of  the 
orbit  the  tendency  of  the  disturbing  force  is  the  same  (511),  and  therefore  while 
the  moon  is  moving  from  N'  to  L'  the  inclination  will  diminish,  and  while  it  is 
moving  from  L'  to  N  the  inclination  will  increase.  The  diminutions  and  incre- 
ments will  compensate  each  other,  and  the  original  inclination  will  be  regained  at 
the  close  of  the  revolution. 

When  the  nodes  are  in  syzigies  there  will  be  no  change  of  inclination  (510). 

In  the  situations  of  the  sun  represented  hi  Figs.  116  and  117,  the  inclination 
will  decrease  from  q  to  L  and  from  q  to  L',  and  increase  from  L  to  q  and  from  L' 
to  q ;  the  effects  being  the  same  as  when  the  nodes  are  in  quadratures  over  the 
arcs  qL  and  LJT  hi  Fig.  116,  and  NL  and  Lg'  in  Fig.  117,  and  being  reversed  over 


FiQ.  116. 


Fio.  117. 


the  arcs  Ng  and  TS'q  in  Fig.  116,  and  qN  and  q"K'  in  Fig.  117.  When  the  sun  hag 
the  position  represented  in  Fig.  116,  the  arcs  of  increase  Lq'  and  L'q  will  be 
greater  than  the  arcs  of  diminution  qL  and  q'L'.  The  disturbing  force  will  also  be 
greater  in  the  former  arcs  than  in  the  latter.  In  the  position  supposed,  therefore, 
there  will  be,  on  the  whole,  an  increase  of  inclination  every  revolution.  When  the 
sun  is  in  the  position  represented  in  Fig.  117,  the  arcs  of  diminution  qL  and  q"L' 
will  be  the  greater ;  and  the  force  in  them  will  also  be  the  greater.  In  this  case, 
therefore,  there  will  be  a  diminution  of  the  inclination  each  revolution  of  the 
moon. 

When  the  sun  is  on  the  other  side  of  the  line  of  nodes,  the  results  will  be  the 
same  as  in  the  positions  diametrically  opposite. 

514.  Consequences  of  the  sun's  motion  during  the  revolution  of  the  moon.  As 
the  sun  moves  from  S  towards  N'  (Fig.  116)  the  arcs  Lq',  L'g,  over  which  there  is 
an  increase  of  the  inclination,  will  increase ;  and  the  arcs  qL,  g'L',  over  which 
there  Ls  a  diminution,  will  diminish.  The  motion  of  the  sun  will,  therefore,  in  ap- 


296      PERTURBATIONS   OF  ELLIPTIC  MOTION  OF  THE  MOON. 

preaching  the  descending  node,  render  the  increase  of  the  inclination  each  revolu- 
tion of  the  moon  greater  than  it  otherwise  would  be.  When  the  sun  is  receding 
from  the  ascending  node,  the  corresponding  arcs  will  experience  corresponding 
changes,  and  therefore  the  diminution  will  now  be  less  than  if  the  sun  were  sta- 
tionary. 

The  results  will  be  similar  for  the  opposite  quadrants  on  the  other  side  of  the 
line  of  nodes 

515.  Epochs  of  greatest  and  least  Inclinations.      Since  the  inclination  dimi- 
nishes as  the   sun  recedes   from  either  node,  and   increases   as  it  approaches 
either  node,  it  will  be  the  least  when  the  nodes  are  in  quadratures,  and  the  greatest 
when  they  are  in  syzigies. 

It  is  important  to  observe  that  the  change  of  inclination  which  we  have  been 
considering  is  modified  by  the  retrograde  motion  of  the  node ;  and  thus,  that,  be- 
sides the  variations  of  this  element  connected  with  the  motions  of  the  moon  and 
sun,  there  is  another  extending  through  the  period  employed  by  the  node  in  com- 
pleting a  revolution  with  respect  to  both  the  sun  and  moon. 

516.  Perturbations  Periodic.     The  perturbations  of  the  elliptic  motion  of 
the  moon,  comprising  inequalities  of  orbit  longitude,  and  variations  in  the  form 
and  position  of  the  orbit,  which  have  now  been  under  consideration,  depend  upon 
the  configurations  of  the  sun  and  moon,  with  respect  to  each  other,  the  perigee  of 
each  orbit,  and  the  node  of  the  moon's  orbit.    Their  effects  will  disappear  when  the 
configurations  upon  which  they  depend  become  the  same.    They  are  therefore 
periodical. 

517.  The  Perturbations  of  the  Motions  of  a  Planet,  produced  by 
the  action  of  another  planet,  are  precisely  analogous  to  the  perturbations  of  the 
motions  of  the  moon,  produced  by  the  action  of  the  sun.     The  disturbing  forces 
are  obviously  of  the  same  kind,  and  they  are  subject  to  variations  from  precisely 
similar  causes.    But,  owing  to  the  smallness  of  the  masses  of  the  planets  and 
their  great  distances,  their  disturbing  forces  are  much  more  minute  than  the  dis- 
turbing force  of  the  sun.    From  this  cause,  together  with  the  slow  relative  motion 
of  the  disturbing  and  disturbed  body,  the  motion  of  the  apsides  and  nodes,  and 
the  accompanying  variations  of  eccentricity  and  inclination,  are  very  much  more 
gradual  in  the  case  of  the  planets  than  in  the  case  of  the  moon.    Their  periods  com- 
prise many  thousands  of  years,  and  on  this  account  they  are  called  Secular  Mo- 
tions or  Variations.     In  consequence  of  the  greater  feebleness  of  the  disturbing 
forces,  the  periodical  inequalities  are  also  much  less  in  amount.     Moreover,  as  the 
motion  of  a  planet  is  much  slower  than  that  of  the  moon,  and  as  the  variations  of 
its  orbit  are  more  gradual  than  those  of  the  lunar  orbit,  the  compensations  pro- 
duced by  a  change  of  configurations  are  much  more  slowly  effected,  and  thus  the 
periods  of  the  inequalities  are  much  longer. 

518.  Acceleration  of  the  Moon.     The  motions  of  the  moon  would  be 
subject  to  no  secular  variations  if  the  apparent  orbit  of  the  sun  were  unchange- 
able; but  the  secular  variation  of  the  eccentricity  of  the  sun's  orbit,  which  an- 
swers to  an  equal  variation  of  the  eccentricity  of  the  earth's  orbit,  that  is  produced 
by  the  action  of  the  planets,  gives  rise  to  a  secular  inequality  in  the  motion  of  the 
moon,  called  the  Acceleration  of  the  Moon.    This  inequality  was  discovered  from 
observation.    Its  physical  cause  was  first  made  known  by  Laplace. 


MASSES  AND  DENSITIES  OF    THE   PLANETS.  297 


CHAPTER  XXIV. 

RELATIVE  MASSES  AND  DENSITIES  OF  THE  SUN,  MOON,  AND 
PLANETS:  —  RELATIVE  INTENSITY  OF  THE  FORCE  OF  GRA- 
VITY AT  THEIR  SURFACE. 

519.  Determination  of  the  masses  of  the  Planets,    The 

perturbations  which  a  planet  produces  in  the  motions  of  the 
other  planets,  depend  for  their  amount  chiefly  upon  the  ratio  of 
the  mass  of  the  planet  to  the  mass  of  the  sun,  and  the  ratio  of 
the  distance  of  the  planet  from  the  sun  to  the  distance  of  the 
planet  disturbed  from  the  same  body.  Now,  the  ratio  of  the 
distances  is  known  by  the  methods  of  Spherical  Astronomy; 
consequently,  the  observed  amount  of  the  perturbations  ought 
to  make  known  the  ratio  of  the  masses,  the  only  unknown  ele- 
ment upon  which  it  depends. 

This  is  one  method  of  determining  the  masses  of  the  planets. 
The  masses  of  those  planets  which  have  satellites  may  be  found 
by  another  and  simpler  method,  viz.  :  by  comparing  the  attract- 
ive force  of  the  planet  for  either  one  of  its  satellites  with  the 
attractive  force  of  the  sun  for  the  planet.  These  forces  are  to 
each  other  directly  as  the  masses  of  the  planet  and  sun,  and  in- 
versely as  the  squares  of  the  distances  of  the  satellite  from  the 
primary  and  of  the  primary  from  the  sun.  Thus  calling  the 
forces/  F,  the  masses  ra,  M,  and  the  distances  c?,  D,  we  have 


whence  we  obtain  m  :  M  :  :f<F  :  FDa.  If  we  regard  the  orbits 
as  circles,  then  d  and  D  will  be  the  mean  distances,  respectively, 
of  the  satellite  from  the  primary,  and  of  the  primary  from  the 
sun,  and  are  given  in  Tables  II,  III,  and  YI.  The  ratio  of  /to  F 
is  equal  to  the  ratio  of  the  versed  sines  of  the  arcs  actually  de- 
scribed by  the  satellite  and  primary,  in  some  short  interval  of 
time  ;  since  these  are  sensibly  equal  to  the  distances  that  the  two 
bodies  are  deflected  in  this  interval  from  the  tangents  to  their 
orbits,  towards  the  centres  about  which  they  are  revolving:  and 
since  the  rates  of  motion  and  dimensions  of  the  orbits  of  the 
planet  and  satellite  are  known,  these  arcs  and  their  versed  sines 
are  easily  determined. 

Table  IY  exhibits  the  relative  masses  of  the  sun,  moon,  and 


298      RELATIVE  MASSES  OF  THE   SUN,  MOON,  AND  PLANETS. 

planets,  according  to  the  most  received  determinations,  that  of 
the  sun  being  denoted  by  1. 

520.  Computation  of  the  Densities  of  the  Planets.    The 

quantities  of  matter  of  the  sun,  moon,  and  planets,  as  well  as  their 
bulks,  being  known,  their  densities  may  be  easily  computed ;  for, 
the  densities  of  bodies  are  proportional  to  their  quantities  of 
matter  divided  by  their  bulks. 

Table  IV  contains  the  densities  of  the  sun,  moon,  and  planets, 
that  of  the  earth  being  denoted  by  1.  It  will  be  seen  on  inspect- 
ing it,  that  the  densities  of  the  planets  decrease  from  Mercury  to 
Saturn ;  and  that  the  four  planets  most  distant  from  the  sun 
are  much  less  dense  than  the  four  which  are  nearest  the  sun. 

521.  The  Comparative  Forces  of  Gravity  at  the  surface 
of  the  sun,  moon,  and  planets,  may  also  readily  be  found,  when 
the  masses  and  bulks  of  these  bodies  are  known.     For  suppos- 
ing them  to  be  spherical,  and  not  to  rotate  on  their  axes,  the 
force  of  gravity  at  their  surface  will  be  directly  as  their  masses 
and  inversely  as  the  squares  of  their  radii,  or,  in  other  words, 
proportional  to  their  masses  divided  by  the  squares  of  their  radii. 
The  centrifugal  force  at  the  surface  of  a  planet,  generated  by  its 
rotation  on  its  axis,  diminishes  the  gravity  due  to  the  attraction 
of  the  matter  of  the  planet.     The  diminution  thus  produced  on 
any  of  the  planets  is  not,  however,  very  considerable.     The 
method  of  determining  the  centrifugal  force  at  the  surface  of  a 
body  in  rotation,  is  given  in  treatises  on  Mechanics.     (See  Table 


18s 


216' 


101. 


Fia.  no. 


Fia.  112. 


FORM  AND  DENSITY  OF  THE  EARTH.         299 


CHAPTER  XXV. 

FORM  AND  DENSITY  OF  THE  EARTH  : — CHANGES  OF  ITS  PERIOD 
OF    KOTATION. —  PRECESSION    OF    THE    EQUINOXES,   AND 

NUTATION. 

522.  WE  have  already  seen  (105)  that  measurements  made 
upon  the  earth's  surface  establish  that  the  figure  of  the  earth  is 
that  of  an  oblate  spheroid,  and  that  the  oblateness  at  the  poles 
is  jd^ 

523.  Deiifity  of  the  Earth.     From  the  amount   and   law 
of  variation  of  the  force  of  gravity  upon  the  earth's  surface, 
ascertained  by  observations  upon  the  length  of  the  seconds'  pen- 
dulum, it  is  proved  that  the  matter  of  the  earth  is  not  homo- 
geneous, but  denser  towards  the  centre,  and  that  it  is  arranged 
in  concentric  strata  of  nearly  an  elliptical  form   and   uniform 
density. 

The  fact  of  the  greater  density  of  the  earth  towards  its  centre 
has  also  been  established  by  observations  upon  the  deviation  of 
a  plumb-line  from  the  vertical,  produced  by  the  attraction  of  a 
mountain ;  the  amount  of  the  deviation  being  ascertained  by 
observing  the  difference  in  the  zenith  distances  of  the  same  star, 
as  measured  with  a  zenith-sector  on  opposite  sides  of  the  moun- 
tain. To  the  north  of  the  mountain  the  plummet  was  drawn 
towards  the  south,  and  the  zenith  distance  of  a  star  to  the  north 
of  the  zenith  was  diminished ;  while  to  the  south  of  the  moun- 
tain the  plummet  was  drawn  towards  the  north,  and  the  zenith 
distance  of  the  same  star  was  increased  by  an  equal  amount : 
and  thus  the  difference  of  the  two  measured  zenith  distances  was 
equal  to  twice  the  deviation  of  the  plumb-line  from  the  true  ver- 
tical in  either  of  the  positions  of  the  instrument  (allowance 
being  made  for  the  difference  of  latitude  of  the  two  stations,  as 
determined  from  the  distance  between  them  and  the  known 
length  of  a  degree). 

Such  observations  were  made  for  the  purpose  of  determin- 
ing the  mean  density  of  the  earth  by  Dr.  Maskelyne,  in  1774, 
on  the  sides  of  the  mountain  Schehallien  in  Scotland.  The  ob- 
served deviation  of  the  plumb-line  made  known  the  ratio  of  the 
attraction  of  the  mountain  to  that  of  the  whole  earth,  and  thus 
the  relative  quantities  of  matter  in  the  mountain  and  earth. 
These  being  ascertained,  and  the  figure  and  bulk  of  the  moun- 
tain having  been  determined  by  a  survey,  the  relative  density 
of  the  earth  and  mountain  became  known  by  the  principle  men- 


300  FORM   AND  DENSITY  OF  THE   EARTH. 

tioned  in  Art.  520,  and  thence  the  actual  density  of  the  earth  ; 
the  density  of  the  mountain  having  been  found  by  experiment. 
The  result  was,  that  the  mean  density  of  the  earth  is  4.95.  Later 
determinations  make  it  5.44. 

524.  Explanation  of  Spheroidal  Form  of  Earth.    The 
spheroidal  form  of  the  surface  of  the  earth  and  of  its  internal 
strata  is  easily  accounted  for,  if  we  suppose  the  earth  to  have 
been  originally  in  a  fluid  state.     The  tendency  of  the  mutual 
attraction  of  its  particles  would  be  to  give  it  a  spherical  form ; 
but  by  virtue  of  its  rotation,  all  its  particles,  except  those  lying 
immediately  on  the  axis,  would  be  animated  by  a  centrifugal 
force  increasing  with  their  distance  from  the  axis.     If,  therefore, 
we  conceive  of  two  columns  of  fluid  extending  to  the  earth's 
centre,  one  from  near  the  equator,  and  the  other  from  near  either 
pole,  the  weight  of  the  former  would  by  reason  of  the  centrifu- 
gal force  be  less  than  that  of  the  latter.     In  order,  then,  that 
they  may  sustain  each  other  in  equilibrio,  that  near  the  equator 
must  increase  in  length,  and  that  near  the  pole  diminish.     As 
this  would  be  true  at  the  same  time  for  every  pair  of  columns 
situated  as  we  have  supposed,  the  surface  of  the  whole  body  of 
fluid  about  the  poles  must  fall,  and  that  of  the  fluid  about  the 
equator  rise.     In  this  manner  the  earth  would  become  flattened 
at  the  poles  and  protuberant  at  the  equator. 

Upon  a  strict  investigation  it  appears  that  a  homogeneous 
fluid  of  the  same  mean  density  with  the  earth,  and  rotating  on 
its  axis  at  the  same  rate  that  the  earth  does,  would  be  in  equili- 
brium, if  it  had  the  figure  of  an  oblate  spheroid,  of  which  the 
axis  was  to  the  equatorial  diameter  as  229  to  230,  or  of  which 
the  oblateness  was  ^J-g-.  If  the  fluid  mass  supposed  to  rotate  on 
its  axis  be  not  homogeneous,  but  be  composed  of  strata  that 
increase  in  density  from  the  surface  to  the  centre,  the  solid  of 
equilibrium  will  still  be  an  elliptic  spheroid,  but  the  oblateness 
will  be  less  than  when  the  fluid  is  homogeneous. 

525.  Po§sible  Changes    Of   Period    of   Rotation.     The 
time  of  the  earth's  rotation,  as  well  as  the  position  of  its  axis, 
would  change  if  any  variation  should  take  place  in  the  distribu- 
tion of  the  matter  of  the  earth,  or  in  case  of  the  impact  of  a 
foreign  body. 

If  any  portion  of  matter  be,  from  any  cause,  made  to  approach 
the  axis,  its  velocity  will  be  diminished,  and  the  velocity  lost 
being  imparted  to  the  mass,  will  tend  to  accelerate  the  rotation. 
If  any  portion  of  matter  be  made  to  recede  from  the  axis,  the 
opposite  effect  will  be  produced,  or  the  rotation  will  be  retarded. 
In  point  of  fact,  the  changes  that  take  place  in  the  position  of 
the  matter  of  the  earth,  whether  from  the  washing  of  rains  upon 
the  sides  of  mountains,  or  evaporation,  or  any  other  known 
cause,  are  not  sufficient  ever  to  produce  any  sensible  alteration 
in  the  circumstances  of  the  earth's  rotation  on  its  axis. 


EARTH'S  AXIS  INVARIABLE.  301 

526.  Earth's    Dimension  and  Axis  Invariable.     It  is 

ascertained  from  direct  observation,  that  there  has  in  reality  been 
no  perceptible  change  in  the  period  of  the  earth's  rotation  since 
the  time  of  Hipparchus,  120  years  before  the  beginning  of  the 
present  era.  We  may  therefore  conclude,  a,  posteriori,  that  there 
has  been  no  material  change  in  the  form  and  dimensions  of  the 
earth  in  this  interval. 

Were  the  axis  of  the  earth  to  experience  any  change  of  posi- 
tion with  respect  to  the  matter  of  the  earth,  the  latitudes  of  places 
would  be  altered.  A  motion  of  100  feet  might  increase  or  di- 
minish the  latitude  of  a  place  to  the  amount  of  1",  an  angle 
which  can  be  measured  by  modern  instruments.  Now,  in  point 
of  fact,  the  latitudes  of  places  have  not  sensibly  varied  since 
their  first  determination  with  accurate  instruments;  therefore,  in 
this  interval  the  axis  of  the  earth  cannot  have  materially  changed. 
Indeed,  since  the  earth's  surface  and  its  internal  strata  are  ar- 
ranged symmetrically  with  respect  to  the  present  axis  of  rotation, 
it  is  to  be  inferred  that  this  axis  is  the  same  as  that  which  ob- 
tained at  the  epoch  when  the  matter  of  the  earth  changed  from 
a  fluid  to  a  solid  state. 

527.  Physical   Theory  of   Precession  and   Nutation. 
The  motions  of  the  earth's  axis,  along  with  the  whole  body  of 
the  earth,. which  give  rise  to  the  Precession  of  the  Equinoxes 
and  Nutation,  are  consequences  of  the  spheroidal  form  of  the 
earth,  inasmuch  as  they  are  produced  by  the  actions  of  the  sun 
and  moon  upon  that  portion  of  the  matter  of  the  earth  which  lies 
on  the  outside  of  a  sphere  conceived   to  be  described  about 
the  earth's  axis.    The  physical  theory  of  the  phenomena  in  ques- 
tion is  analogous  to  that  of  the  retrogradation  of  the  moon's 
nodes.     The  sun  produces  a  retrograde  movement  of  the  points 
in  which  the  circle  described  by  each  particle  of  the  protuberant 
mass  cuts  the  plane  of  the  ecliptic,  as  it  does  of  the  moon's  nodes ; 
the  effect  produced  is,  however,  exceedingly  small,  by  reason  of 
the  inertia  of  the  interior  spherical   mass  connected  with  the 
external  mass  upon  which  the  action  takes  place.     The  moon, 
in  like  manner,  occasions  a  retrograde  movement  of  the  nodes 
of  the  same  particles  on  the  plane  of  its  orbit.      The  actions 
of  the  sun  and  moon  will  not  be  the  same  each  revolution  of  a 
particle.     That  of  the  sun  will  vary  during  the  year  with  the 
angular  distance  of  the  sun  from  the  node  (510) ;  and  that  of 
the  moon  will  vary  during  each  month  with  the  distance  of  the 
moon  from  the  node,  and  also  during  a  revolution  of  the  nodes 
of  the  moon's  orbit  by  reason  of  the  change  in  the  inclination  of 
the  orbit  to  the  equator.     The  mean  effect  of  both  bodies  is  the 
precession;  the  inequality  resulting  from  the  change  in  the  sun's 
action  during  the  year  is  the  solar  nutation ;  and  the  inequality 
consequent  upon  the  retrogradation  of  the  moon's  nodes  is  the 
lunar  nutation,  or  the  chief  part  of  it. 


THE  TIDES. 


CHAPTER  XXVI. 

THE  TIDES. 

528.  THE  alternate  rise  and  fall  of  the  surface  of  the  ocean 
twice  in  the  course  of  a  lunar  day,  or  about  25  hours,  is  the  phe- 
nomenon known  by  the  name  of  the  Tides.     The  rise  of  the 
water  is  called  the  Flood  Tide,  and  the  fall  the  Ebb  Tide. 

529.  Times  of   High  and  Low  Water.       The   interval 
between  one  high  water  and  the  next  is,  at  a  mean,  half  a  mean 
lunar  day,  or  12h.  25m.     Low  water  has  place  nearly,  but  not 
exactly,  at  the  middle  of  this  interval;  the  tide,  in  general,  em- 
ploying nine  or  ten  minutes  more  in  ebbing  than  in  flowing.    As 
the  interval  between  one  period  of  high  water  and  the  second 
following  one  is  a  lunar  day,  or  Id.  Oh.  50m.,  the  retardation  in 
the  time  of  high  water  from  one  day  to  another  is  50m.,  in  its 
mean  state. 

The  time  of  high  water  is  mainly  dependent  upon  the  position 
of  the  moon,  being  always,  at  any  given  place,  about  the  same 
length  of  time  after  the  moon's  passage  over  the  superior  or  in- 
ferior meridian.  As  to  the  length  of  the  interval  between  the 
two  periods,  at  different  places,  in  the  open  sea  it  is  only  from 
two  to  three  hours;  but  on  the  shores  of  continents,  and  in 
rivers,  where  the  water  meets  with  obstructions,  it  is  very  dif- 
ferent at  different  places,  and  in  some  instances  is  of  such  length 
that  the  time  of  high  water  seems  to  precede  the  moon's  passage. 

530.  Height  of  Tide.      The   height  of  the  tide   at   high 
water  is  not  always  the  same,  but  varies  from  day  to  day ;  and 
these  variations  have  an  evident  relation  to  the  phases  of  the  moon. 
It  is  greatest  soon  after  the  syzigies ;  after  which  it  diminishes 
and  becomes  the  least  soon  after  the  quadratures. 

The  tides  which  occur  near  the  syzigies,  are  called  the  Spring 
Tides  ;  and  those  which  occur  near  the  quadratures  are  called  the 
Neap  Tides. 

The  highest  of  the  spring  tides  is  not  that  which  has  place 
nearest  to  new  or  full  moon,  but  is  in  general  the  third  following 
tide.  In  like  manner  the  lowest  of  the  neap  tides  is  the  third  or 
fourth  tide  after  the  quadrature. 

The  spring  tides  are,  in  general,  from  once  and  a  half  to  twice 
the  height  of  the  neap  tides.  At  Brest,  in  France,  the  former 
rise  to  the  height  of  19.3  feet,  and  the  latter  only  to  9.2  feet. 


PHENOMENA  OF  THE  TIDES.  303 

On  the  Atlantic  coast  of  the  United  States  the  spring  tides  ex- 
ceed the  neap  tides  in  the  proportion  of  3  to  2. 

The  tides  are  also  affected  by  the  decimations  of  the  sun  and  moon: 
thus,  the  highest  spring  tides  in  the  course  of  the  year  are  those 
which  occur  near  the  equinoxes.  The  extraordinarily  high  tides 
which  frequently  occur  at  the  equinoxes  are,  however,  in  part 
attributable  to  the  equinoctial  gales.  Also,  when  the  moon  or 
the  sun  is  out  of  the  equator,  the  evening  and  morning  tides 
differ  somewhat  in  height.  At  Brest,  in  the  syzigies  of  the  sum- 
mer solstice,  the  tides  of  the  morning  of  the  first  and  second  day 
after  the  syzigy  are  smaller  than  those  of  the  evening  by  6.6 
inches.  They  are  greater  by  the  same  quantity  in  the  syzigies 
of  the  winter  solstice. 

The  distance  of  the  moon  from  the  earth  has  also  a  sensible  influ- 
ence upon  the  tides.  In  general,  they  increase  and  diminish  as 
the  distance  diminishes  and  increases,  but  in  a  more  rapid 
ratio. 

531.  Daily  Retardation  of  High  Water.     The  daily  re- 
tardation of  the  time  of  high  water  varies  with  the  phases  of 
the  moon.     It  is  at  its  minimum  towards  the  syzigies,  when  the 
tides  are  at  their  maximum ;  and  at  its  maximum  towards  the 
quadratures,  when  the  tides  are  at  their  minimum.     It  also  varies 
with  the  distances  of  the  sun  and  moon  from  the  earth,  and  with 
their  declinations. 

532.  Physical   Theory  of   the   Tide§.     The  facts  which 
have  been  detailed  indicate  that  the  tides  are  produced  by  the 
actions  of  the  sun  and  moon  upon  the  waters  of  the  ocean ;  but 
in  a  greater  degree  by  the  action  of  the  moon.     To  explain 
them,  let  us  suppose  at  first  that  the  whole  surface  of  the  earth 
is  covered  with  water.     We  remark,  in  the  first  place,  that  it  is 
not  the  whole  attractive  force  of  the  moon  or  sun  which  is 
effective  in  raising  the  waters  of  the  ocean,  but  the  difference  in 
the  actions  of  each  body  upon  the  different  parts  of  the  earth ; 
or,  more  precisely,  that  the  phenomenon  of  the  tides  is  a  conse- 
quence of  the  inequality  and  non-parallelism  of  the  attractive 
forces  exerted  by  the  moon,  as  well  as  by  the  sun,  upon  the  dif- 
ferent particles  of  the  earth's  mass.     From  this  cause  there  re- 
sults a  diminution  in  the  gravity  of  the  particles  of  water  at  the 
surface,  for  a  certain  distance  about  the  point  immediately  under 
the  moon,  and  the  point  diametrically  opposite  to  this,  and  an 
augmentation  for  a  certain  distance  on  the  one  side  and  the  other 
of  the  circle  90°  distant  from  these  points,  or  of  which  they  are 
the  geometrical  poles :  in  consequence  of  which  the  water  falls 
about  this  circle  and  rises  about  these  points.     That  the  actions 
of  the  moon  upon  the  different  parts  of  the  earth's  mass  are 
really  unequal,  is  evident  from  the  fact  that  these  parts  are  at 
different  distances  from  the  moon.     To  show  that  the  inequality 
will  give  rise  to  the  results  just  noted,  let  us  suppose  that  the 


304: 


THE  TIDES. 


circle  acbd  (Fig  118)  represents  the  earth,  and  M  the  place  of 

the  moon  ;  then  a  will  be  the  point  of 
the  earth's  surface  directly  under  the 
moon,  b  the  point  diametrically  oppo- 
site to  this,  and  the  right  line  c?c,  per- 
pendicular to  MO  ,will  represent  the 
circle  traced  on  the  earth's  surface  90° 
distant  from  a  and  I.  Now,  the  at- 
traction of  the  moon  for  the  general 
mass  of  the  earth  is  the  same  as  if  the 
whole  mass  were  concentrated  at  the 
centre  0.  But  the  centre  of  the  earth 
is  more  distant  from  the  moon  than 
the  point  a  at  the  surface.  It  follows, 
therefore,  that  a  particle  of  matter  situ- 
ated at  a  will  be  drawn  towards  the 
moon  with  a  proportionally  greater 
force  than  the  centre,  or  than  the  gen- 
eral mass  of  the  earth.  Its  gravity  or 
tendency  towards  the  earth's  centre  will 
therefore  be  diminished  by  the  amount 
of  this  excess.  On  the  other  hand,  the 
centre  is  nearer  to  the  moon  than  the 
point  b.  It  is  therefore  attracted  more 
strongly  than  a  particle  at  b.  The  ex- 
cess will  be  a  force  tending  to  draw 
the  centre  away  from  the  particle  ;  and  the  effect  will  be  the 
same  as  if  the  particle  were  drawn  away  from  the  centre  by  the 
same  force  acting  in  the  opposite  direction.  The  result  then  is, 
that  this  particle  has  its  gravity  towards  the  earth's  centre  dimin- 
ished, as  well  as  the  particle  at  a.  If  now  we  consider  a  particle 
at  some  point  t  near  to  a,  the  moon's  action  upon  it  (tr)  may  be 
considered  as  taking  effect  partially  in  the  direction  ik  parallel 
to  OM,  and  partially  in  the  direction  of  the  tangent  or  horizontal 
line  ts.  The  component  (is)  in  the  latter  direction,  will  have  no 
tendency  to  alter  the  gravity  of  the  particle  towards  the  earth's 
centre.  The  component  (sr)  in  the  direction  tk,  will  obviously 
be  less  than  the  actual  force  of  attraction  tr]  and  the  difference 
will  be  greater  in  proportion  as  the  particle  is  more  remote  from 
a.  This  component  will  decrease  gradually  from  a,  where  it  is 
equal  to  the  attractive  force,  while  the  attraction  for  the  centre 
is  less  than  for  a  by  a  certain  finite  difference  :  it  is  plain,  there- 
fore, that  the  component  in  question  will  be  greater  than  the 
attraction  for  the  centre,  in  the  vicinity  of  the  point  #,  and  for  a 
certain  distance  from  it  in  all  directions.  The  gravity  of  the  par- 
ticles will  therefore  be  diminished  for  a  certain  distance  from  this 
point.  In  a  similar  manner  it  may  be  shown  that  it  will  also  be 
diminished  for  a  certain  distance  from  the  point  b.  Let  us  now 


FlG  llg 


THEORY  OF  THE  TIDES.  305 

consider  a  particle  at  c,  90°  from  the  points  a  and  b.  The  at- 
traction of  the  moon  for  it  will  take  effect  in  the  two  directions  cl 
andcO.  The  force  in  the  latter  direction  alone  will  alter  the 
gravity  of  the  particle  ;  and  this,  it  is  plain,  will  increase  it. 
The  same  effect  will  extend  to  a  certain  distance  from  c  in  both 
directions. 

A  strict  mathematical  investigation  would  show  that  the 
gravity  is  diminished  for  a  distance  of  55°  from  a  and  b  in  all 
directions;  and  is  augumented  for  a  distance  of  35°  on  each  side 
of  the  circle  cfc,  90°  distant  from  the  points  a  and  b.  These 
distances  are  represented  in  the  Figure. 

This  may  be  easily  made  out  by  means  of  the  expression  for  the  radial  disturbing 

force  of  the  sun  in  its  action  upon  the  moon  (505),  viz.  —  x  y  (1  —  3  cos2  0).   If  we 

aa 

consider  ra  as  denoting  the  mass  of  the  moon,  a  the  moon's  distance  from  the  earth's 
centre,  y  the  distance  of  a  particle  of  matter  at  some  point  t  of  the  earth's  surface 
from  the  earth's  centre,  and  </>  the  angular  distance  or  elongation  (M0<)  of  the  same 
particle  from  the  moon,  as  seen  from  the  centre  of  the  earth,  it  will  express  the 
change  in  the  gravity  of  a  particle  at  the  earth's  surface,  produced  by  the  moon'8 
action.  The  points  a  and  b  will  answer  to  conjunction  and  opposition,  and  the  points 
c  and  d  to  the  quadratures.  Now  we  have  already  seen  (505)  that  the  gravity  of  the 
moon  is  increased  at  the  quadratures,  and  for  35°  on  each  side  of  them  ;  and  dimin- 
ished at  the  syzigies,  and  55°  from  them  in  both  directions.  It  follows,  therefore, 
that  the  same  is  true  for  particles  of  matter  at  the  earth's  surface. 

In  consequence  of  the  earth's  diurnal  rotation,  the  parts  of  the 
surface,  at  which  the  rise  and  fall  of  the  water  will  take  place, 
will  be  continually  changing.  Were  the  entire  rise  and  fall  pro- 
duced instantaneously,  the  points  of  highest  water  would  con- 
stantly be  the  precise  points  in  which  the  line  of  the  centres  of 
the  moon  and  earth  intersects  the  surface,  and  it  would  always  be 
high  water  on  the  meridian  passing  through  these  points,  both 
in  the  hemisphere  where  the  moon  is,  and  in  the  opposite  one. 
On  the  west  side  of  this  meridian,  the  tide  would  be  flowing;  on 
the  east  side  of  it,  it  would  be  ebbing  ;  and  on  the  meridian 
at  right  angles  to  the  same,  it  would  be  low  water.  But  it  is 
plain  that  the  effects  of  the  moon's  action  would  not  be  instan- 
taneously produced,  and  therefore  that  the  points  of  highest  water  I  y-  - 
will  fall  behind  the  moon.  "^p 

533.  Comparative  Effect§  of  Sun  and  jTKoon.  It  is 
evident  that  the  sun  will  produce  precisely  similar  effects  with 
the  moon,  and  will  raise  a  tide  wave  similar  to  the  lunar  tide 
wave,  which  will  follow  it  in  its  diurnal  motion. 

To  show  that  the  effects  of  the  sun  are  less  in  degree  than  those  of  the  moon,  let 
us  take  the  general  expression  for  the  change  of  the  moon's  gravity,  arising  from 
the  action  of  the  sun,  namely, 


.  .  .(a). 

From  what  we  have  seen  in  the  previous  article,  this  formula  will  serve  to  ex- 
press the  change  in  the  gravity  of  a  particle  of  matter  upon  the  earth's  surface, 
produced  by  the  sun's  action,  if  we  take  m  =  the  mass  of  the  sun,  as  before,  a 
rr  its  distance  expressed  in  terms  of  the  radius  of  the  earth  as  unity,  y  =  the  dia- 

20 


306  THE   TIDES. 

tance  of  the  particle  from  the  centre  of  the  earth,  and  <t>  =  its  elongation  from  the 
sun,  as  seen  from  the  earth's  centre.  If  we  designate  the  corresponding  quantities 
for  the  moon  by  m'  a'  y,  0,  we  shall  have  for  the  change  of  the  gravity  of  a  particle, 
produced  by  the  moon's  action, 

j£  x  y  (i  _  3  cos2  f  )  .  .  .  (6). 

For  particles  at  equal  elongations  from  the  sun  and  moon,  we  shall  have  9  the 
game  in  expressions  (a)  and  (6),  and  y  may  be  regarded  as  the  same  without  material 
error.  For  such  particles,  then,  the  alterations  of  the  gravity,  produced  by  the  sun 

and   moon,  will  bear  the  same  ratio  to  each  other  as  the  quantities  —  and  —• 

a3  a" 

Now,  if  we  give  to  m,  m'.  a,  a',  their  values,  we  shall  find  that  the  latter  quantity 
is  about  2^-  times  greater  than  the  former.  Accordingly,  the  effect  of  the  moon's 
action,  at  corresponding  elongations  of  the  particles,  and  therefore  generally,  is 
about  2£  times  greater  than  that  of  the  sun. 

534.  Combined  Effects  of  Sun  and  Moon.  The  actual 
tide  will  be  produced  by  the  joint  action  of  the  sun  and  moon, 
or  it  may  be  regarded  as  the  result  of  the  combination  of  the 
lunar  and  solar  tide  waves. 

At  the  time  of  the  syzigies,  the  action  of  the  sun  and  moon 
will  be  combined  in  producing  the  tides,  both  bodies  tending  to 
produce  high  as  well  as  low  water  at  the  same  places.  But  at 
the  quadratures  they  will  be  in  opposition  to  each  other,  the  one 
tending  to  raise  the  surface  of  the  water  where  the  other  tends  to 
depress  it,  and  vice  versti.  The  tides  should,  therefore,  be  much 
higher  at  the  syzigies  than  at  the  quadratures. 

Between  the  syzigies  and  the  quadratures  the  two  bodies  will 
neither  directly  conspire  with  each  other,  nor  directly  oppose 
each  other,  and  tides  of  intermediate  height  will  have  place.  The 
points  of  highest  water  will  also,  in  the  configuration  supposed, 
neither  be  the  vertices  of  the  lunar  nor  of  the  solar  tide  wave, 
but  certain  points  between  them.  This  circumstance  will  occa- 
sion a  variation  in  the  length  of  the  interval  between  the  time 
of  the  moon's  passage  and  the  time  of  high  water. 

Spring  and  Neap  Tides.  The  effect  of  the  moon's  action  being 
to  that  of  the  sun's  nearly  as  2-J  to  1  (533),  the  spring  tides  will 
be  to  the  neap  tides  nearly  as  2-J-  to  1.  For,  let  x  —  the  effect  of 
the  moon,  and  y  =  the  effect  of  the  sun  :  then  the  ratio  of  x  -f  y 
to  x  —  y  will  be  the  ratio  of  the  heights  of  the  spring  and  neap 
tides.  Now 


x  =  2.34y,  and  thus  5-±^-=    '  y  =  2.5. 

x  —  y       2.34?/  —  y 

"We  have  already  seen  that  the  ratio  obtained  from  observation  is 
less  than  this. 

The  height  of  the  joint  tide,  as  well  as  the  interval  between 
the  time  of  high  water  and  that  of  the  moon's  meridian  passage, 
will  vary  not  only  with  the  elongation  of  the  moon  from  the  sun, 
but  also  with  the  distance  and  declination  of  the  moon.  For, 


COMPARISON  OF  THEORY  WITH   OBSERVATION.  307 

expressions  (a)  and  (b)  above  given,  show  that  the  intensities  of  the 
moon's  and  sun's  actions  vary  inversely  as  the  cube  of  their  dis- 
tance; and  the  changes  of  the  declinations  of  the  two  bodies 
must  be  attended  with  a  change  both  in  the  absolute  and  rela- 
tive situation  of  the  vertices  of  the  lunar  and  solar  tide-waves. 


COMPARISON  OF  THE  THEORY  OF  THE  TIDES  WITH  THE 
RESULTS  OF  OBSERVATION. 

535.  The  laws  of  the  tides,  which  should  obtain  on  the  hypo- 
thesis of  the  earth  being  entirely  covered  with  water,  are  found  to 
correspond  only  partially  with  those  of  the  actual  tides.      The 
continents  have  a  material  influence  upon  the  formation  and 
propagation  of  the  tide-wave.      The  actual  phenomena  of  the 
tides  have  been  carefully  observed,  for  many  years,  at  numerous 
points  along  the  coast  lines  of  continents   and   on   the  shores 
of  islands :  and  the  results  of  the  observations  have  been  sub- 
jected to  a  thorough  discussion  by  several  distinguished  astrono- 
mers and  physicists.     As  one  result  of  the  discussion  the  deter- 
mination has  been  effected  of  a  system  of  Cotidal  Lines  ;  that  is, 
a  set  of  lines  connecting  those  places  at  which  high  tide  occurs  at 
the  same  instant,  from  hour  to  hour.     A  chart  has  been  con- 
structed showing,  at  intervals  of  lh,  2h,  3h,  &c.,  after  the  meridian 
transit  of  the  moon  at  Greenwich,  the  cotidal  lines  of  the  South- 
ern, Atlantic,  and  Pacific  Oceans.    These  lines  show  the  varying 
form  of  the  ridge  of  the  tide-wave  as  it  proceeds  on  its  course, 
and  by  the  greater  or  less  distance  between  them  the  rate  of 
propagation  of  the  wave  in  different  oceans  and   in   different 
parts  of  the  same  ocean.     Along  the  coasts  they  are,  for  the 
most  part,  constructed  from  actual  observations,  but  their  exten- 
sions into  the  open  sea  are  mostly  inferential. 

536.  Tide-wave  of   the  Atlantic   Ocean.     By  examin- 
ing the  chart  of  cotidal  lines  we  learn  that  the  floodtide  of  the 
Atlantic  Ocean  is,  for  the  most  part,  produced  by  a  derivative 
tide-wave,  sent  off  from  the  great  wave  which,  in  the  Southern 
Ocean,  follows  the  moon  in  its  diurnal  motion  around  the  earth. 
At  6  hours  after  the  meridian  transit  of  the  moon  at  Greenwich, 
the   derivative   tide-wave   stretches  from   the   coast  of  Upper 
Guinea  to  the  coast  of  Brazil,  a  little  to  the  south  of  the  narrow- 
est part  of  the  Atlantic.     Three  hours  later  it  has  advanced,  by 
estimation,  in  mid-ocean,  to  about  24°  of  north  latitude ;  and 
in  3  hours  more,  or  12  hours  after  the  meridian  transit  of  the 
moon  at  Greenwich,  it  has  reached  the  Atlantic  coast  of  the 
United  States.     It  advances  more  rapidly  in  the  open  sea  than 
along  the  coasts,  where  the  depth  of  the  water  is  less.    It  is  there- 
fore convex  towards  the  north.  Thus,  at  the  hour  just  mentioned, 
it  stretches  nearly  parallel  to  the  general  trend  of  our  Atlantic 


303  THE   TIDES. 

coast,  along  its  whole  extent  into  the  northern  Atlantic,  and 
there  curves  around  to  the  south-east,  so  as  to  strike,  at  its  east- 
ern end,  the  N.  W.  coast  of  Africa  (lat.  23°).  The  same  wave 
does  not  reach  the  coast  of  Spain  until  more  than  two  hours 
later. 

537.  Nature  and  Velocity  of  tlie  Tide-wave.  The  tidal- 
wave  is  of  the  nature  of  a  wave  of  translation.  In  this  form  of 
wave  there  is  no  oscillation  proper ;  but  the  particles  of  the  fluid, 
in  a  cross  section  perpendicular  to  the  line  of  propagation,  by  the 
transit  of  the  wave  are  raised,  transferred  forward,  and  brought 
to  rest  in  the  direction  of  the  motion  in  a  new  place ;  with  the  same 
extent  of  transference  of  each  particle  throughout  the  whole  depth 
of  the  wave.  Whereas  in  ordinary  oscillatory  waves,  such  as  those 
caused  by  the  wind,  the  individual  particles  oscillate  in  vertical 
circles,  or  ellipses,  and  return  to  their  original  position.  A  wave 
of  translation  travels  with  a  velocity  equal  to  that  acquired  by 
a  heavy  body  in  falling  freely  by  gravity  through  a  height  equal 
to  half  the  mean  depth  of  the  fluid,  reckoned  from  the  top  of  the 
wave  to  the  bottom  of  the  channel.  Its  velocity  is  therefore  di- 
rectly proportional  to  the  square  root  of  the  depth  of  the  fluid. 
The  rate  of  propagation  of  an  oscillatory  wave,  on  the  other 
hand,  is  independent  of  the  depth,  and  varies  only  with  the 
breadth  of  the  wave. 

The  moon  tends  to  draw  the  wave  which  it  raises  along  with 
it  in  its  diurnal  course,  at  the  rate  of  1,000  miles  per  hour  at  the 
equator  ;  but  it  appears  that  the  tidal  wave  actually  travels  at  a 
much  less  rapid  rate.  Setting  out  from  the  Eastern  Pacific, 
where  it  lags  about  2  hours  behind  the  moon,  it  travels  westward 
in  about  12  hours  to  New  Zealand.  From  thence  to  the  Cape 
of  Good  Hope,  passing  south  of  Australia,  it  occupies  17  hours, 
and  has  an  average  velocity  of  about  470  miles  per  hour.  From 
the  Cape  of  Good  Hope  the  portion  of  the  wave  that  passes 
northward  into  the  Atlantic  traverses  the  distance  to  the  coasl 
of  the  United  States  in  about  11  hours;  which  is  at  the  average 
rate  of  565  miles  per  hour.  The  tide-wave  accordingly  does 
not  reach  our  Atlantic  coast  until  about  40  hours  after  it  origin 
ated  in  the  South-eastern  Pacific.  The  average  velocity  of  56£ 
miles  in  the  South  and  North  Atlantic,  answers  to  a  depth  of 
21,500  feet,  or  about  4  miles.  The  average  velocity  in  mid 
ocean  is  greater  than  this,  and  answers  to  a  greater  depth. 

The  velocity  of  the  tide-wave  becomes  rapidly  reduced  aftei 
the  wave  strikes  the  shallow  waters  of  the  coast,  to  100  mile: 
per  hour;  or  even  less  than  50  miles  per  hour  in  bays  and  sounds 
As  a  necessary  consequence  the  breadth  of  the  wave  diminishei 
with  its  velocity.  At  a  velocity  of  565  miles  per  hour  it  has  j 
breadth  of  7,000  miles.  When  the  velocity  is  reduced  to  10( 
miles  per  hour  the  breadth  is  only  1,240  miles. 


TIDES  OF  THE  ATLANTIC  COAST.  309 


TIDES  OF  THE  ATLANTIC  COAST  OF  THE  UNITED  STATES. 

538.  General  Phenomena.     The  phenomena  of  the  tides 
as  they  occur  along  the  entire  coast  line  of  the  United  States,  have 
been  carefully  deduced  by  the  late  Superintendent  of  the  Coast 
Survey,   from  the  systematic  tidal  observations   carried  on  in 
connection  with  the"  Survey.     The  following  are  the  more  im- 
portant general  results  obtained  from  the  discussion: 

1.  The  cotidul  lines,  in  the  vicinity  of  the  Atlantic  coast,  are 
nearly  parallel  to  the  general  trend  of  the  coast.     The  ridge  of 
the  tide-wave,  as  it  approaches  the  coast,  is  therefore  nearly  par- 
allel to  the  coast  line.     This  wave,  when  it  reaches  the  most 
prominent  points  of  the  coast,  has  a  mean  height  of  about  2  feet 
above  the  lowest  point  of  the  ebb-wave,  or  mean  low-water 
level. 

2.  The  coast  is  physically  divided,  by  projecting  headlands, 
into  three  great  bays,  each  of  which  has  its  particular  system  of 
cotidal  lines,  running  nearly  parallel  to  the  shore.  These  bays  may 
be  designated  as  the  Southern,  Middle,  and  Eastern  Bays.     The 
Southern  Bay  lies  between  Cape  Florida  and  Cape  Hatteras ;  the 
Middle  Bay  between  Cape  Hatteras  and  Nan  tucket  (eastern  end) ; 
and  the  Eastern  Bay  between  Nan  tucket  and  Cape  Sable  (Nova 
Scotia).     The  latter  is  supposed  to  be  a  portion  of  a  greater  bay, 
from  Nantucket  to  Cape  Race  (Newfoundland).     In  the  Southern 
Bay,  the  mean  rise  and  fall,  or  range  of  the  tides  along  the 
shores,  increases  from  about  2  feet  at  the  capes  to  7  feet  at  Port 
Eoyal,  at  the  head  of  the  bay.     In  the  Middle  Bay,  the  range 
increases  from  2  feet  to  nearly  5  feet  at  Sandy  Hook  and  Cape 
May.     In  the  Eastern  Bay  the  tides  are  more  complex,  owing  to 
greater  irregularities  in  the  shore  line,  and  the  influence  of  shoals. 
The  heights  increase  rapidly  from  Nantucket  to  Cape  Cod ;  the 
mean  range  being  2  feet  at  Nantucket  and  9.2  feet  at  Province- 
town.     At  Cape  Ann  (the  northern  cape  of  Massachusetts  Bay) 
it  is  about  the  same.     From  Cape  Ann  north  ward  to  Portsmouth 
there  is  a  decrease  of  about  half  a  foot  in  the  mean  range  of  the 
tides.     From  thence,  following  the  shore  line  towards  the  north- 
east, it  increases  at  an  augmenting  rate  until  at  the  entrance  of 
the  Bay  of  Fundy  the  tide  rises,  on  the  average,  18  feet  above 
Jow  water. 

539.  Tides  of  Inner  Bays.     The  tides   of  Delaware  Bay, 
New  York  Bay,  and  Narragansett  and  Buzzard  Bays,  present,  on 
a  smaller  scale,  the  same  phenomena  of  increase  in  the  height  of 
the  tides  in  ascending,  as  the  three  great  bays,  or  undulations  of 
the  coast.     On  the  contrary,  in  Chesapeake  Bay,  which  widens 
and  changes  direction  at  right  angles  immediately  from  the  en- 
trance, the  tides  diminish  in  height,  as  a  general  rule,  in  going 
up  the  bay. 


310  THE  TIDES. 

The  tide-wave,  on  entering  Massachusetts  Bay,  increases  some- 
what, viz.,  from  9  feet  above  low  water  at  the  entrance  to  10 
feet  at  Plymouth  and  Boston. 

In  the  Bay  of  Fundy,  the  tides  rise  to  a  much  greater  height 
than  on  any  other  part  of  the  Atlantic  Coast.  At  St.  Johns,  N. 
B.,  the  mean  rise  and  fall  of  the  tide  is  19.3  feet;  and  at  Shad- 
wood  Point,  at  the  head  of  the  bay,  no  less  than  36  feet.  The 
ordinary  spring  tides  attain,  at  the  latter  place,  to  the  height  of 
50  feet.  Special  tides  have  been  known  to  rise  20  feet  higher. 
This  remarkable  accumulation  of  the  tidal  waters  results  from 
the  great  contraction  in  the  width  of  the  bay  or  channel  into 
which  the  ascending  wave  is  forced. 

45O.  Tides  of  Channels.  In  channels  peculiar  tides  occur, 
in  consequence  of  the  meeting  of  the  waves  which  enter  the  chan- 
nels at  their  two  extremities.  Where  the  two  flood  waves  meet 
in  the  same  state,  a  tide  equal  to  the  sum  of  their  two  heights  is 
produced  by  their  superposition.  At  other  points  the  tides  are 
variously  modified  by  the  interference  of  the  waves. 

Tides  in  Sounds  present  similar  peculiarities. 

541.  Tides  of  L,ong  Island  Sound.  The  great  tidal  wave  from  the  At- 
lantic enters  the  Sound  between  Point  Judith  and  Montauk  Point ;  and  another 
portion  of  this  wave  enters  New  York  Bay,  and  passing  through  Hell  Gate,  meets 
the  wave  propagated  through  the  Sound  from  the  eastward.  The  point  of  meet- 
ing of  the  crests  of  the  two  waves  is  off  Sands'  Point,  at  the  head  of  the  Sound. 
At  Montauk  Point  the  mean  height  of  the  tide-wave,  above  low  water,  is  2  feet, 
and  at  Sandy  Hook  4.8  feet.  At  Sands'  Point  it  is  7.7  feet;  exceeding  the  sum  of 
the  heights  at  the  two  entrances  by  nearly  I  foot,  owing  to  the  narrowing  of  the 
Sound.  The  mean  range  of  the  tide  declines  in  both  directions  from  Sands'  Point. 
At  Bridgeport  it  is  6.6  feet;  at  New  Haven  5.8  feet;  at  New  London  and  Stonington 
between  2£  feet,  and  2$  feet;  and  at  Point  Judith  3  feet. 

The  tide  is  propagated  from  Montauk  Point  to  the  head  of  the  Sound  in  3  hours. 
It  travels  from  Fisher's  Island  to  Sands'  Point,  95  miles,  in  2h.  1m. ;  or  at  the 
average  rate  of  47  miles  per  hour.  This  agrees  approximately  with  the  velocity 
as  theoretically  computed  from  the  soundings  taken  by  the  Coast  Survey,  accord- 
ing to  the  law  of  propagation  of  a  wave  of  translation  (537).  At  Fisher's  Island 
it  is  about  60  miles,  and  becomes  reduced  to  30  miles  at  the  head  of  the  Sound, 
where  the  depth  of  the  water  is  less.  In  the  East  Eiver  the  rate  of  propagation 
of  the  tide  is  only  about  7£  miles  per  hour. 

Owing  to  the  retardation  of  the  tide- wave  in  the  shallow  waters  near  the  Con- 
necticut shore,  it  is  nearly  parallel  to  the  shore,  from  the  head  of  the  Sound  to  a 
distance  of  some  20  miles  east  of  New  Haven  harbor.  Accordingly  high  water 
occurs  at  about  the  same  hour  along  this  extent  of  the  shore.  Farther  to  the  east, 
the  line  of  the  tide- wave  is  inclined  to  the  shore  line,  and  the  tides  occur  earlier. 

542.  Tidal  Currents.  The  currents  produced  by  the  tides  in  the  shallow  waters 
of  bays,  sounds,  and  rivers,  are  not  to  be  confounded  with  the  transmission  of 
the  tide-wave.  Their  velocity  is  but  a  few  miles  per  hour ;  and  the  turn  of  the 
current,  or  tide-stream,  does  not  in  general  correspond  to  the  turn  of  the  tido, 
and  may  occur  at  quite  a  different  hour.  For  example,  at  Montauk  Point  the  ebb- 
stream  does  not  begin  until  half  ebb-tide,  and  in  New  York  Bay  it  begins  at  one- 
sixth  of  the  ebb  tide. 

Tidal  currents  owe  their  origin  to  the  resistance  opposed  by  shallow  waters, 
and  contracted  channels,  to  the  free  propagation  of  the  tide- wave,  and  to  differences 
of  hydrostatic  level.  They  have  the  greatest  velocity  in  narrow  channels,  as  in 
the  Race  off  Fisher's  Sound,  and  in  Hell  Gate.  About  the  time  of  the  turn  of  the 
tide,  at  the  head  of  the  Sound,  there  is  a  certain  interval  of  slackwater  there.  After 
the  tide- wave  begins  to  move  in  the  opposite  direction,  the  accumulative  effect  of 


TIDES  OF  THE   PACIFIC  COAST.  311 

the  resistances  determines,  in  a  certain  interval  of  time,  a  sensible  current,  which 
shows  itself  first  at  the  surface  and  in-shore,  but  soon  becomes  general.  In  mid- 
channel,  throughout  the  Sound,  the  outward  motion  of  the  water  commences  shortly 
after  high  water  at  the  head  of  the  Sound,  and  evidently  depends  upon  it 

A  similar,  but  still  more  striking  fact  is  observed  in  the  Irish  Channel.  The 
turn  of  the  stream,  whether  flood  or  ebb,  is  simultaneous  throughout  the  entire 
length  of  the  channel.  It  is  coincident  with  the  time  of  high  or  low  water  at 
Morecambe  Bay,  north  of  Liverpool,  where  the  tides  coming  round  the  extremities 
of  Ireland  finally  meet.  The  times  of  slackwater  throughout  the  channel,  therefore, 
correspond  with  the  times  of  high  and  low  water  at  Morecambe  Bay.  In  the 
Irish  Channel  there  are  two  spots,  in  one  of  which  the  stream  runs  with  considera- 
ble velocity  without  the  tide  either  rising  or  falling,  while  in  the  other  the  water 
rises  and  falls  from  sixteen  to  twenty  feet  without  having  any  visible  horizontal 
motion  at  its  surface. 

The  average  maximum  drift  of  the  current  in  Long  Island  Sound,  is  2.2  knots  per 
hour.  The  average  maximum  current  velocity  opposite  the  west  end  of  Fisher's  Is- 
land is  nearly  4  J  knots  per  hour ;  and  at  Hell  Gate  nearly  6  knots.  In  New  York  har- 
bor it  is  3.7  knot?,  and  in  the  Bay  3  knots.  The  point  of  meeting  of  the  two  flood 
streams  in  the  East  River,  is  a  little  to  the  east  of  Throgs'  Neck.  To  the  east  and 
west  of  that  point,  both  the  flood  and  ebb  streams  run  in  opposite  directions. 

The  mean  duration  of  the  flood  stream  at  different  points  of  Long  Island  Sound 
varies  between  4j  hours  and  7|  hours.  The  corresponding  limits  for  the  ebb 
stream  are  5h.  and  8^h.  The  mean  duration  of  slackwater  varies  between  Om. 
and  45m.  It  is  at  most  places  less  than  10m.  The  duration  of  the  ebb  or  flood 
stream,  differs  as  much  as  £  of  an  hour  in  successive  tides ;  but  commonly  not 
more  than  10m.  The  set  of  the  currents  is  ordinarily  nearly  parallel  to  the 
shore. 

543.  Tides  of  Rivers.  The  tide-wave  that  enters  the  mouth 
of  a  river  is  propagated  according  to  the  same  laws  as  a  wave 
that  comes  in  at  the  entrance  of  a  sound,  or  channel.  The  ve- 
locity varies  with  the  depth  of  water ;  and  the  height  of  the  tide 
increases  where  the  river  contracts,  and  decreases  where  it 
widens.  Thus,  in  a  tidal  river  of  considerable  length,  the  tide 
may  have  various  heights  at  different  points.  The  ascending  flood 
tide  may  also  be  encountered  by  the  descending  ebb  tide.  On  the 
Hudson  the  tide  rises  at  West  Point,  55  miles  from  New  York, 
2.7  feet ;  at  Tivoli,  nearly  100  miles  from  New  York,  4  feet ; 
and  at  Albany,  2.3  feet. 

In  the  shallow  parts  of  rivers,  the  tide-wave  becomes  converted 
into  a  tidal  current,  by  which  alone  the  tide  is  transmitted.  In 
rivers  the  duration  of  the  ebb  tide  is  considerably  longer  than 
that  of  the  flood.  Thus,  at  Philadelphia  and  Richmond,  the 
ebb  continues  2|-  hours  longer  than  the  flood  tide. 


TIDES  OF  THE  PACIFIC  COAST. 

544.  Cot  id  al  Lines.     The  cotidal  lines  of  the  Pacific  coast 
of  the  United  States  are  approximately  parallel  to  the  coast. 
Thus,  high  tide  occurs  at  about  the  same  hour  from  San  Fran- 
cisco to  Vancouver's  Island.     South  of  San  Francisco  the  tide- 
wave  arrives  at  an  earlier  hour;  at  the  southern  extremity  of 
California,  about  2J  hours  earlier. 

545.  Diurnal  Inequality.     The  tides  of  the  Pacific  coast 


312  THE  TIDES. 

are  remarkable  for  the  great  inequality  that  -prevails  between  the 
heights  of  tivo  successive  tides,  as  measured  from  the  high  water 
mark  of  each  tide  to  the  next  succeeding  low  water  mark.  The 
difference  of  level  of  the  two  successive  high  tides  is  less  con- 
spicuous, but  quite  marked.  The  differences  are  greater  for  the 
ebb  than  for  the  flood  tides.  These  diurnal  inequalities  increase 
with  the  moon's  declination,  north  or  south;  and  vanish  en- 
tirely when  the  moon  is  in  the  equator.  When  the  moon's  de- 
clination is  north,  the  highest  of  the  two  high  tides  of  the 
twenty-four  hours  occurs  at  San  Francisco  about  11  ^  hours  after 
the  moon's  superior  transit ;  and  when  the  declination  is  south, 
the  lowest  of  the  two  high  tides  occurs  about  this  interval  after 
the  transit.  When  the  moon  has  its  greatest  declination  the 
mean  range  of  the  highest  tide  is  nearly  7  feet,  and  of  the  low- 
est tide  from  1-J-  ft.  to  3  ft.  The  lowest  tide  sometimes  amounts 
to  only  two  or  three  inches. 

According  to  Professor  Bache,  the  tides  that  occur  on  the 
western  coast,  near  the  maximum  of  the  moon's  declination  and 
for  several  days  on  each  side  of  it,  result  from  the  interference 
of  a  semi-diurnal  and  diurnal  wave,  which  at  the  maximum  of 
each  are  nearly  equal  in  magnitude,  the  crest  of  the  diurnal 
wave  being  at  that  period  about  eight  hours  in  advance  of  that 
of  the  semi-diurnal  wave.  This  diurnal  wave  exists  only  when 
the  moon  has  a  considerable  declination. 

On  the  Atlantic  coast  the  corresponding  inequality  at  the  time 
of  the  moon's  greatest  declination,  is  a  small  fraction  of  the 
height  of  the  tide,  and  is  generally  not  more  than  one  foot.  A 
similar  remark  may  be  made  of  the  tides  of  the  coast  of  Europe. 


TIDES  OF  THE  GULF  OF  MEXICO. 

546.  On  the  northern  coast  of  the  Gulf  of  Mexico,  from 
Florida  westward,  there  is  but  one  tide  in  the  24  hours,  and  the 
mean  range  of  this  tide  is  only  from  1  foot  to  1-J  feet.  The 
second  tide  is  doubtless  obliterated  by  the  interference  of  the 
semi-diurnal  flood-tide  with  a  diurnal  ebb-tide ;  as  happens  ap- 
proximately on  the  Pacific  coast  (545).  For  some  three  to  five 
days,  about  the  time  when  the  moon  is  crossing  the  equator, 
when  the  diurnal  inequality  should  vanish,  from  the  absence  of 
the  diurnal  wave  (545),  there  are  generally  two  tides  at  the  same 
places  on  the  coast,  the  rise  and  fall  being  quite  small.  The 
greatest  rise  and  fall  of  the  single  day-tide  occurs  when  the 
moon's  declination  is  the  greatest. 

The  small  height  of  the  tides  in  the  Gulf  of  Mexico  is  at- 
tributable chiefly  to  the  fact  that  the  width  of  the  gulf  is  three 
or  four  times  greater  than  that  of  the  two  channels  through  which 
the  tide-wave  enters  it. 


TIDES  OF  THE  COAST  OF  EUROPE.  313 

TIDES  OF  THE  MEDITERRANEAN. 

547.  The  average  height  of  the  tide  in  the  Mediterranean  is 
said  not  to  exceed  1£  feet,  though  at  some  ports,  as  Tunis  and 
Venice,  it  sometimes  amounts  to  3  or  4  feet.     The  Mediterranean 
is  of  sufficient  extent  for  the  sun  and  moon  to  produce  a  sensible 
tide  by  their  direct  action.     A  derivative  tide-wave,  from  the 
Atlantic  Ocean,  should  also  enter  the  Straits  of  Gibraltar,  and 
spread  out  laterally  as  it  advances ;  but  the  ebb  and  flow  from 
this  cause  is  said  to  be  slight. 

TIDES  OF  INLAND  SEAS  AND  LAKES. 

548.  Lakes  and  inland  seas  have  no  perceptible  tides,  or  only 
very  small  tides,  for  the  reason  that  their  extent  is  not  sufficient 
to  admit  of  any  sensible  inequality  of  gravity,  as  the  result  of 
the  action  of  the  moon  (532).     A  tide  of  nearly  2  inches  has 
been  detected  at  Chicago,  on  the  southwestern  shore  of  Lake 
Michigan. 

TIDES  OF  THE  COAST  OF  EUROPE. 

549.  The  tide- wave  advancing  from  the  south,  makes  a  con- 
siderable angle  with  the  coast  of  Europe,  and  thus  the  tide  occurs 
continually  later  in  following  the  coast  from  the  Straits  of  Gibral- 
tar northward ;  and  along  its  entire  extent  from  two  to  twelve 
hours  later  than  the  corresponding  tides  on   the  coast  of  the 
United  States.     Similar  varieties  of  tidal  phenomena  occur  on 
either  coast. 

The  highest  tides  prevail  in  the  Bristol  Channel,  and  the  Bay 
of  St.  Malo,  on  the  northwest  coast  of  France.  At  the  head  of 
the  Bristol  Channel,  and  of  the  Bay  of  St.  Malo,  the  spring  tides 
sometimes  rise  to  the  height  of  50  feet.  The  mean  range  of 
spring  tides  is  26  feet  at  Liverpool,  nearly  13  feet  at  Portsmouth, 
and  about  20  feet  at  London  Docks.  On  the  coast  of  France,  the 
height  of  the  tides  at  different  ports  falls  approximately  between 
the  same  limits  as  on  the  coast  of  England. 

The  lowest  tides  occur  on  the  eastern  coast  of  Ireland,  to  the 
north  of  the  entrance  to  St.  George's  Channel.  At  Courtown, 
about  30  miles  north  of  Tuskar,  there  is  scarcely  any  rise  or  fall 
of  the  water.  From  that  point  the  height  of  the  tide  increases 
about  equally  in  every  direction,  from  0  to  15  feet  on  the 
opposite  coast.  The  remarkably  low  tides  at  that  locality  result 
from  the  fact  that  the  tide  stream  is  diverted  by  a  promontory 
at  the  entrance  of  the  channel  to  the  opposite  shore. 

ESTABLISHMENT  OF  THE  PORT.— TIDE-TABLES. 

550.  The  interval  between  the  time  of  the  moon's  crossing  the 


314  THE  TIDES. 

meridian  and  the  time  of  high  water  at  a  given  place  is  nearly 
constant.  It  varies  only  between  moderate  assignable  limits. 
The  mean  interval  on  the  days  of  new  and  full  moon  is  called 
the  establishment  of  the  port.  The  average  of  the  intervals  dur- 
ing a  month's  tides,  is  called  the  mean,  or  correct  establishment. 
The  mean  establishment  of  Boston  is  lib.  27m. ;  of  New  Haven 
llh.  16rn.;  of  New  York  8h.  13m.;  of  Charleston,  S.  C.,  7h. 
and  26rn. ;  of  San  Francisco  12h.  12m. 

551.  Calculation  of  Time  of  High  Water.  When  the 
mean  establishment  of  a  port  is  known,  the  time  of  high  water 
on  any  day  may  be  approximately  determined.  The  hour  of 
transit  of  the  moon  on  the  given  day  is  to  be  taken  from  the 
Nautical  Almanac  and  added  to  the  mean  establishment ;  the 
result  will  be  the  time  of  high  water.  If  the  time  thus  determined 
falls  in  the  succeeding  day,  half  a  lunar  day  (12h.  25m.)  is  to  be 
subtracted,  as  this  is  the  mean  interval  between  two  successive 
tides. 

On  the  day  of  new  or  full  moon,  the  time  of  the  next  high 
water  after  noon,  will  be  approximately  equal  to  the  establish- 
ment of  the  port. 

In  the  annual  Coast  Survey  Reports  a  table  is  published,  giving  the  interval  be- 
tween the  time  of  the  moon's  transit  and  the  time  of  high  water  for  different  hours 
of  transit,  and  for  the  principal  ports  on  the  U.  S.  coast.  If  the  time  of  the  moon's 
transit  on  any  day  be  obtained  from  the  Nautical  Almanac,  the  interval  correspond- 
ing to  this  time  in  the  table,  added  to  the  time  of  transit,  will  give  more  accurately 
the  time  of  high  water. 

552.  A  tide  table  for  the  coast  of  the  United  States,  is  published  in  the  same 
Reports,  giving  for  numerous  points  of  the  coast  the  mean  values  of  the  interval  be- 
tween the  time  of  the  moon's  transit  and  time  of  high  water,  the  rise  and  fall  of  the 
tides,  the  rise  and  fall  of  the  spring  and  neap  tides,  the  duration  of  flood  and  of 
ebb  tide,  and  the  duration  of  the  stand,  or  the  period  of  time  during  which  the  sur- 
face of  the  water  neither  rises  nor  falls.  A  table  is  also  given  showing,  for  various 
ports,  the  rise  and  fall  of  tides  corresponding  to  different  hours  of  the  moon's  transit ; 
from  which,  by  taking  the  time  of  transit  for  any  day  from  the  Almanac,  the  cor- 
responding  rise  and  fall  of  the  tide  may  be  obtained  for  any  of  the  ports  mentioned 
in  the  tabla 


PART  HI. 

ASTRONOMICAL    PROBLEMS. 


EXPLANATIONS  OF  THE  TABLES. 

THE  Tables  which  form  a  part  of  this  work,  and  which  are  em- 
ployed in  the  resolution  of  the  following  Problems,  consist  of  Ta- 
bles of  the  Sun,  Tables  of  the  Moon,  Tables  of  the  Mean  Places 
of  some  of  the  Fixed  Stars,  Tables  of  Corrections  for  Refraction, 
Aberration, and  Nutation,  and  Auxiliary  Tables. 

The  Tables  of  the  Sun,  which  are  from  XVII  to  XXXIV,  in- 
elusive,  are,  for  the  most  part,  abridged  from  Delambre's  Solar  Ta- 
bles. The  mean  longitudes  of  the  sun  and  of  his  perigee  for  the 
beginning  of  each  year,  found  in  Table  XVIII,  have  been  com- 
puted from  the  formulae  of  Prof.  Bessel,  given  in  the  Nautical  Al- 
manac of  1837.  The  Table  of  the  Equation  of  Time  was  reduced 
from  the  table  in  the  Connaissance  des  Terns  of  1810,  which  is 
more  accurate  than  Delambre's  Table,  this  being  in  some  instances 
liable  to  an  error  of  2  seconds.  The  Table  of  Nutation  (Table 
XXVII)  was  extracted  from  Francceur's  Practical  Astronomy. 
The  maximum  of  nutation  of  obliquity  is  taken  at  9". 25.  The 
Tables  of  the  Sun  will  give  the  sun's  longitude  within  a  frac- 
tion of  a  second  of  the  result  obtained  immediately  from  De- 
lambre's Tables,  as  corrected  by  Bessel.  The  Tables  of  the 
Moon,  which  are  from  XXXIV  to  LXXXV,  inclusive,  are 
abridged  and  computed  from  Burckhardt's  Tables  of  the  Moon. 
To  facilitate  the  determination  of  the  hourly  motions  in  longi- 
tude and  latitude,  the  equations  of  the  hourly  motions  have  all 
been  rendered  positive,  like  those  of  the  longitude.  Some  few  new 
tables  have  been  computed  for  the  same  purpose.  The  longitude 
and  hourly  motion  Si  longitude  will  very  rarely  differ  from  the  re 
suits  of  Burckhardt's  Tables  more  than  0".5,  and  never  as  much 
as  !'• .  The  error  of  the  latitude  and  hourly  motion  in  latitude  will 
be  still  less.  The  other  tables  have  been  taken  from  some  of  the 
most  approved  modern  Astronomical  Works.  (For  the  principles 
of  the  construction  of  the  Tables,  see  Note  1.,  Appendix.) 

Before  entering  upon  the  explanation  of  each  of  the  tables,  it 
will  be  proper  to  define  a  few  terms  that  will  be  made  use  of  in  the 
sequel. 

The  given  quantity  with  which  a  quantity  is  taken  from  a  tablej 
is  called  the  A  rgument  of  this  quantity. 


Table  I,  contains  the  latitudes  and  longitudes  from  the  meridian 
of  Greenwich,  of  various  conspicuous  places  in  different  parts  of 
the  earth.  The  longitudes  serve  to  make  known  the  time  at  any 
one  of  the  places  in  the  table,  when  that  at  any  of  the  others  is 
given.  The  latitude  of  a  place  is  an  important  element  in  various 
astronomical  calculations. 

Table  II,  is  a  table  of  the  Elements  of  the  Orbits  of  the  Planets 
with  their  secular  variations,  which  serve  to  make  known  the  ele- 
ments at  any  given  epoch  different  from  that  of  the  table.  From 
these  the  elliptic  places  of  the  planets  at  the  given  epoch  may  be 
computed.  Table  III.,  is  a  similar  table  for  the  Moon. 

Table  II.  (a)  gives  the  mean  distances,  &c.,  of  the  Planetoids. 

Tables  IV,  V,  VI,  VII,  require  no  explanation. 

Table  VIII,  gives  the  mean  Astronomical  Refractions  ;  that  is, 
the  refractions  which  have  place  when  the  barometer  stands  at  30 
inches,  and  the  thermometer  of  Fahrenheit  at  50°. 

Table  IX,  contains  the  corrections  of  the  Mean  Refractions  for 
+  1  inch  in  the  barometer,  and —  1°  in  the  thermometer,  from 
which  the  corrections  to  be  applied,  at  any  observed  height  of  the 
barometer  and  thermometer,  are  easily  derived. 

Table  X,  gives  the  Parallax  of  the  Sun  for  any  given  altitude  on 
a  given  day  of  the  year  ;  for  reducing  a  solar  observation  made  at 
the  surface  of  the  earth  to  what  it  would  have  been,  if  made  at  the 
centre. 

Table  XI,  is  designed  to  make  known  the  Sun's  Semi-diurnal 
Arc,  answering  to  any  given  latitude  and  to  any  given  declination 
of  the  s~un ;  and  thus  the  time  of  the  sun's  rising  and  setting,  and 
the  length  of  the  day. 

Table  XII,  serves  to  make  known  the  value  of  the  Equation  of 
rl  .'me,  with  its  essential  sign,  which  is  to  be  applied  to  the  apparent 
time  to  convert  it  into  the  mean.  If  the  sign  of  the  equation  taken 
from  the  table  be  changed,  it  will  serve  for  the  conversion  of  mean 
time  into  apparent.  This  table  is  constructed  for  the  year  1840. 

Table  XIII,  is  to  be  used  in  connection  with  Table  XII,  when 
the  given  date  is  in  any  other  year  than  1840.  It  furnishes  the 
Secular  Variation  of  the  Equation  of  Time,  from  which  the  pro- 
portional part  of  its  variation  in  the  interval  between  the  given  date 
and  the  epoch  of  Table  XII  is  easily  derived. 


EXPLANATION  OF  THE  TABLES.  317 

Table  XIV,  contains  certain  other  Corrections  to  be  applied  tr 
the  equation  of  time  taken  from  Table  XII,  when  its  exact  value 
to  within  a  small  fraction  of  a  second,  is  desired. 

Table  XV,  gives  the  Fraction  of  the  Year  corresponding  to  each 
date.  This  table  is  useful  when  quantities  vary  by  known  and  uni- 
form degrees,  in  deducing  their  values  at  any  assumed  time  from 
their  values  at  any  other  time. 

Table  XVI,  is  for  converting  Hours,  Minutes,  and  Seconds  into 
decimal  parts  of  a  Day. 

Table  XVII,  is  for  converting  Minutes  and  Seconds  of  a  degree 
into  the  decimal  division  of  the  same.  It  will  also  serve  for  the 
conversion  of  minutes  and  seconds  of  time  into  decimal  parts  of  an 
hour. 

The  last  two  tables  will  be  found  frequently  useful  in  arithmeti- 
cal operations 

Table  XVIII,  is  a  table  of  Epochs  of  the  Sun's  Mean  Longi- 
tude, of  the  Longitude  of  the  Perigee,  and  of  the  Arguments  for 
finding  the  small  equations  of  the  Sun's  place.  They  are  all  cal- 
culated for  the  first  of  January  of  each  year,  at  mean  noon  on  the 
meridian  of  Greenwich.  Argument  I.  is  the  mean  longitude  of  the 
Moon  minus  that  of  the  Sun ;  Argument  II.  is  the  heliocentric 
longitude  of  the  Earth ;  Argument  III.  is  the  heliocentric  longi- 
tude of  Venus  ;  Argument  IV.  is  the  heliocentric  longitude  of 
Mars ;  Argument  V.  is  the  heliocentric  longitude  of  Jupiter ,  Ar- 
gument VI.  is  the  mean  anomaly  of  the  Moon  ;  Argument  VII.  ia 
the  heliocentric  longitude  of  Saturn ;  and  Argument  N  is  the  sup- 
plement of  the  longitude  of  the  Moon's  Ascending  Node.  Argu- 
ment I.  is  for  the  first  part  of  the  equation  depending  on  the  action 
of  the  Moon.  Arguments  I.  and  VI.  are  the  arguments  for  the  re- 
maining part  of  the  lunar  equation.  Arguments  II.  and  III.  are  for 
the  equation  depending  on  the  action  of  Venus ;  Arguments  II. 
and  IV.  for  the  equation  depending  on  the  action  of  Mars ;  Argu- 
ments II.  and  V.  for  the  equation  depending  on  the  action  of  Ju- 
piter ;  and  Arguments  II.  and  VII.  for  the  equation  depending  on 
the  action  of  Saturn.  Argument  N  is  the  argument  for  the  Nuta- 
tion in  longitude  :  it  is  also  the  argument  for  the  Nutation  hi  right 
ascension,  and  of  the  obliquity  of  the  ecliptic. 

Table  XIX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
variations  of  the  arguments,  in  the  interval  between  the  beginning 
of  the  year  and  the  first  of  each  month. 

Table  XX,  shows  the  Motions  of  the  Sun  and  Perigee,  and  the 
rariations  of  the  arguments  from  the  beginning  of  any  month  to  the 
beginning  of  any  day  of  the  month  ;  also  the  same  for  Hours. 

Table  XXI,  gives  the  Sun's  Motions  for  Minutes  and  Seconds. 
Tables  XVIII  to  XXI,  inclusive,  make  known  the  mean  longitude 
of  the  Sun  from  the  mean  equinox,  at  any  moment  of  time. 

Table  XXII,  Mean  Obliquity  of  the  Ecliptic  for  the  beginning 


318  ASTRONOMICAL  PROBLEMS. 

of  each  year  contained  in  the  table.  It  is  found  for  any  interme- 
diate time  by  simple  proportion. 

Tables  XXIII,  and  XXIV,  furnish  the  Sun's  Hourly  Motion 
and  Semi-diameter. 

Table  XXV,  is  designed  to  make  known  the  Equation  of  the 
Sun's  Centre.  When  the  equati  m  has  the  negative  sign,  its  sup- 
plement to  12s.  is  given  :  this  is-,  to  be  added  along  with  the  other 
equations  of  longitude,  and  12s.  are  to  be  subtracted  from  the  sum. 

The  numbers  in  the  table  are  the  values  of  the  equation  of  the 
centre,  or  of  its  supplement,  diminished  by  46". 1.  This  constant 
is  subtracted  from  each  value,  to  balance  the  different  quantities 
added  to  the  other  equations  of  the  longitude,  in  order  to  render 
them  affirmative.  The  epoch  of  this  table  is  the  year  1840. 

Table  XXVI,  gives  the  Secular  Variation  of  the  Equation  of  the 
Sun's  Centre,  from  which  the  proportional  part  of  the  variation  in 
the  interval  between  the  given  date  and  the  year  1840,  may  be 
derived. 

Table  XXVII,  is  for  the  Nutation  in  Longitude,  Nutation  in 
Right  Ascension,  and  Nutation  of  the  Obliquity  of  the  Ecliptic. 
The  nutation  in  longitude  and  nutation  in  right  ascension,  serve  to 
transfer  the  origin  of  the  longitude  and  right  ascension  from  the 
mean  to  the  true  equinox.  And  the  nutation  of  obliquity  serves  to 
change  the  mean  into  the  true  obliquity. 

Tables  XXVIII  to  XXXIII,  inclusive,  give  the  Equations  of 
the  Sun's  Longitude,  due  respectively  to  the  attractions  of  the 
Moon,  Venus,  Jupiter,  Mars,  and  Saturn. 

Table  XXXIV,  is  for  the  variable  part  of  the  Sun's  Aberration. 
The  numbers  have  all  been  rendered  positive  by  the  addition  of 
the  constant  0".3. 

Table  XXXV,  contains  the  Epochs  of  the  Moon's  Mean  Longi- 
tude, and  of  the  Arguments  of  the  equations  used  in  determining 
the  True  Longitude  and  Latitude  of  the  Moon.  They  are  all  cal- 
culated for  the  first  of  January  of  each  year,  at  mean  noon  on  the 
meridian  of  Greenwich.  The  Argument  for  the  Evection  is  di- 
minished by  30' ;  the  Anomaly  by  2°  ;  the  Argument  for  the  Va- 
riation by  9°,  and  the  mean  longitude  by  9°  45' ;  and  the  Supple- 
ment of  the  Node  is  increased  by  7'.  This  is  done  to  balance  the 
quantities  which  are  added  to  the  different  equations  in  order  to 
render  them  affirmative. 

Tables  XXXVI  to  XL,  inclusive,  give  the  Motions  of  the  Moon, 
and  the  variations  of  the  arguments,  for  Months,  Days,  Hours, 
Minutes,  and  Seconds  ;  and,  together  with  Table  XXXV,  are  for 
finding  the  Moon's  Mean  Longitude  and  the  Arguments,  at  any 
assumed  moment  of  time. 

Tables  XLI  to  LIII,  inclusive,  give  the  various  Equations  of 
the  Moon's  Longitude.  It  is  to  be  observed  with  respect  to  Table 
XLI,  that  the  right  hand  figure  of  the  argument  is  supposed  to  be 
dropped.  But  when  the  greatest  attainable  accuracy  is  desired,  it 


EXPLANATION  OF  THE  TABLES.  319 

can  be  retained,  and  a  cipher  conceived  to  be  written  after  the 
numbers  in  the  columns  of  Arguments  in  the  table.  In  Tables 
L,  LI,  LII,  and  L  V,  the  degrees  will  be  found  by  referring  to  the 
head  or  foot  of  the  column.  (See  Problem  II.,  note  2.) 

Table  LIV  is  for  the  Nutation  of  the  Moon's  Longitude. 

Tables  LV  to  LIX,  inclusive,  are  for  finding  the  Latitude  of 
the  Moon. 

Tables  LX  to  LXIII,  inclusive,  are  for  the  Equatorial  Paral 
lax  of  the  Moon. 

Table  LXIV  furnishes  the  Reductions  of  Parallax  and  of  the 
Latitude  of  a  Place.  The  reduction  of  parallax  is  for  obtaining 
the  parallax  at  any  given  place  from  the  equatorial  parallax.  The 
reduction  of  latitude  is  foi  reducing  the  true  latitude  of  a  place,  as 
determined  by  observation,  to  the  corresponding  latitude  on  the 
supposition  of  the  earth  being  a  sphere.  The  elhpticity  to  which 
the  numbers  in  the  table  correspond  is  3^. 

Tables  LXV  and  LXVI,  Moon's  Semi-diameter,  and  the  Aug- 
mentation of  the  Semi-diameter  depending  on  the  altitude. 

Tables  LXVII  to  LXXXV,  inclusive,  are  for  finding  the 
Hourly  Motions  of  the  Moon  in  Longitude  and  Latitude. 

Table  LXXXVI,  Mean  New  Moons,  and  the  Arguments  for  the 
Equations  for  New  and  Full  Moon,  in  January.  The  time  of 
mean  new  moon  in  January  of  each  year  has  been  diminished  by 
15  hours,  the  sum  of  the  quantities  which  have  been  added  to  the 
equations  in  Table  LXXXIX.  Thus,  4h.  20m.  has  been  added 
to  equation  I. ;  lOh.  10m.  to  equation  II.  ;  10m.  to  equation  III.; 
and  20m.  to  equation  IV. 

Tables  LXXXVII  and  LXXXVIII,  are  used  with  the  preced- 
ing in  finding  the  Approximate  Time  of  Mean  New  or  Full  Moon 
in  any  given  month  of  the  year. 

Table  LXXXIX  furnishes  the  Equations  for  finding  the  Ap- 
proximate Time  of  New  or  Full  Moon. 

Table  XC  contains  the  Mean  Right  Ascensions  and  Declina- 
tions of  50  principal  Fixed  Stars,  for  the  beginning  of  the  year 
1840,  with  their  Annual  Variations. 

Table  XCI  is  for  finding  the  Aberration  and  Nutation  of  the 
Stars  in  the  preceding  catalogue. 

Table  XCII  contains  the  Mean  Longitudes  and  Latitudes  of 
some  of  the  principal  Fixed  Stars,  for  the  beginning  of  the  yeaf 
1840,  with  their  Annual  Variations. 

Tables  XCIII,  XCIV,  XCV,  Second,  Third,  and  Fourth 
Differences.  These  tables  are  given  to  facilitate  the  determina- 
tion, from  the  Nautical  Almanac,  of  the  moon's  longitude  or  lati- 
tude for  any  time  between  noon  and  midnight. 

Table  XCVI,  Logistical  Logarithms.  This  table  is  convenient 
in  working  proportions,  when  the  terms  are  minutes  and  seconds, 
or  degrees  and  minutes,  or  hours  and  minutes, — especially  when 
the  first  term  is  Ih.  or  60m 


320  ASTRONOMICAL  PROBLEMS. 

To  find  the  logistical  logarithm  of  a  number  composed  of  min 
utes  and  seconds,  or  degrees  and  minutes,  of  an  arc ;  or  of  min- 
utes and  seconds,  or  hours  and  minutes,  of  time. 

1 .  If  the  number  consists  of  minutes  and  seconds,  at  the  top  of 
the  table  seek  for  the  minutes,  and  in  the  same  column  opposite 
the  seconds  in  the  left-hand  column  will  be  found  the  logistical 
logarithm. 

2.  If  the  number  is  composed  of  hours  and  minutes,  the  hours 
must  be  used  as  if  they  were  minutes,  and  the  minutes  as  if  they 
were  seconds. 

3.  If  the  number  is  composed  of  degrees  and  minutes,  the  de- 
grees must  be  used  as  if  they  were  minutes,  and  the  minutes  as  if 
they  were  seconds. 

To  find  the  logistical  logarithm  of  a  number  less  than  3600. 

Seek  in  the  second  line  of  the  table  from  the  top  the  number 
next  less  than  the  given  number,  and  the  remainder,  or  the  com- 
plement to  the  given  number,  in  the  first  column  on  the  left :  then 
in  the  column  of  the  first  number,  and  opposite  the  complement, 
will  be  found  the  logistical  logarithm  of  the  sum.  Thus,  to  ob- 
tain the  logarithm  of  1531,  we  seek  for  the  column  of  1500,  and 
opposite  31  we  find  3713. 


PROBLEM  I. 

To  work,  by  logistical  logarithms,  a  proportion  the  terms  of  which 
are  degrees  and  minutes,  or  minutes  and  seconds,  of  an  arc ;  or 
hours  and  minutes,  or  minutes  and  seconds,  of  time. 

With  the  degrees  or  minutes  at  the  top,  and  minutes  or  seconds 
at  the  side,  or  if  a  term  consists  of  hours  and  minutes,  or  minutes 
and  seconds,  with  the  hours  or  minutes  at  the  top,  and  minutes 
or  seconds  at  the  side,  take  from  Table  XCVI.  the  logistical  loga- 
rithms of  the  three  given  terms  ;  add  together  the  logistical  loga- 
rithms of  the  second  and  third  terms  and  the  arithmetical  comple- 
ment of  that  of  the  first  term,  rejecting  10  from  the  index.*  The 
result  will  be  the  logistical  logarithm  of  the  fourth  term,  with 
which  take  it  from  the  table. 

Note  1.  The  logistical  logarithm  of  60'  is  0. 

Note  2.  If  the  second  or  third  term  contains  tenths  of  seconds, 
(or  tenths  of  minutes,  when  it  consists  of  degrees  and  minutes,) 
and  is  less  than  6',  or  6°,  multiply  it  by  10,  and  employ  the  loga- 
rithm of  the  product  in  place  of  that  of  the  term  itself.  The 

*  instead  of  adding  the  arithmetical  complement  of  the  ogarithm  of  the  first 
term,  the  logarithm  itself  may  be  subtracted  from  the  sum  of  the  logarithms  of  the 
other  two  terms. 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.         321 

result  obtained  by  the  table,  divided  by  10,  will  be  the  fourth  term 
of  the  proportion,  and  will  be  exact  to  tenths. 

Note  3.  If  none  of  the  terms  contain  tenths  of  minutes  or  sec- 
onds, and  it  is  desired  to  obtain  a  result  exact  to  tenths,  dimmish 
the  index  of  the  logistical  logarithm  of  the  fourth  term  by  1,  and 
cut  off  the  right-hand  figure  of  the  number  found  from  the  table, 
for  tenths. 

Exam.  1.  When  the  moon's  hourly  motion  is  30'  12",  what  is 
its  motion  in  16m.  24s.  ? 

As  60m.       .  .         0 

:    30'  12"   .  .  2981 

: :    16m.  24s.  .         .  5633 

:    8'  15"      ....  8614 

2.  If  the  moon's  declination  change  1°  31'  in  12  hours,  what 
will  be  the  change  in  7h.  42m.  ? 

As  12h.    .         .         .  ar.  co.  9.3010 

:    1°  31'          .         .         .     1.5973 

•:    7h.42m.      .         .         .        8917 

:    0°  58'         ...    1.7900 

3.  When  the  moon's  hourly  motion  in  latitude  is  2'  26  '.8,  what 
is  its  motion  in  36m.  22s.  ? 

2'  26".8 
60 


As  60m.  .    .    0 
1468   .    .   :  1468"  .    .  3896 
:  :  36m.  22s.   .  2174 

:  890"     .         .  6070 

Ans.  1'  29",0. 

4.  When  the  sun's  hourly  motion  in  longitude  is  2'  28",  what 
is  its  motion  in  49m.  11s.  ?  Ans.  2;  1". 

5.  If  the  sun's  decimation  change  16'  33"  in  24  hours,  what 
will  be  the  change  in  14h.  18m.?  Ans.  9;  52''. 

6.  If  the  moon's  declination  change  54".7  in  one  hour,  what  wrti 
be  the  change  in  52m.  18s.  ?  Ans.  47".7. 


PROBLEM  II. 

To  take  from  a  table  the  quantity  corresponding  to  a  given  value 
of  the  argument,  or  to  given  values  of  the  arguments  of  the 
table 

21 


322  ASTRONOMICAL  PROBLEMS. 

Case  1.  When  quantities  are  given  in  the  table  for  each  sign 
and  degree  of  the  argument. 

With  the  signs  of  the  given  argument  at  the  top  or  bottom,  and 
the  degrees  at  the  side,  (at  the  left  side,  if  the  signs  are  found  at 
the  top;  at  the  right  side,  if  they  are  found  at  the  bottom,)  take  out 
the  corresponding  quantity.  Also  take  the  difference  between  this 
quantity  and  the  next  following  one  in  the  table,  and  say,  60' :  this 
difference  :  :  odd  minutes  and  seconds  of  given  argument :  a  fourth 
term.  This  fourth  term,  added  to  the  quantity  taken  out,  when  the 
quantities  in  the  table  are  increasing,  but  subtracted  when  they  are 
decreasing,  will  give  the  required  quantity. 

Note  1.  When  the  quantities  change  but  little  from  degree  to 
degree  of  the  argument,  the  required  quantity  may  often  be  esti- 
mated, without  the  trouble  of  stating  a  proportion. 

Note  2.  In  some  of  the  tables  the  degrees  or  signs  of  the  quan- 
tity sought,  are  to  be  had  by  referring  to  the  head  or  foot  of  the  col- 
umn in  which  the  minutes  and  seconds  are  found.  (See  Tables 
L,  LI,  LII,  and  LV.)  The  degrees  there  found  are  to  be  taken, 
if  no  horizontal  mark  intervenes ;  otherwise,  they  are  to  be  in- 
creased or  diminished  by  1°,  or  2°,  according  as  one  or  two  marks 
intervene.  They  are  to  be  increased,  or  diminished,  according  as 
their  number  is  less  or  greater  than  the  number  of  degrees  at  the 
other  end  of  the  column. 

Note  3.  If,  as  is  the  case  with  some  of  the  tables,  the  quantities 
in  the  table  have  an  algebraic  sign  prefixed  to  them,  neglect  the 
consideration  of  the  sign  in  determining  the  correction  to  be  applied 
to  the  quantity  first  taken  out,  and  proceed  according  to  the  rule 
above  given.  The  result  will  have  the  sign  of  the  quantity  first 
taken  out.  It  is  to  be  observed,  however,  that  if  the  two  consecu- 
tive quantities  chance  to  have  opposite  signs,  their  numerical  sum 
is  to  be  taken  instead  of  their  difference ;  also  that  the  quantity 
sought  will,  in  every  such .  instance,  be  the  numerical  difference 
between  the  correction  and  the  quantity  first  taken  out,  and,  ac- 
cording as  the  correction  is  less  or  greater  than  this  quantity,  is  to 
be  affected  with  the  same  or  the  opposite  sign. 

Exam.  1.  Given  the  argument  7s-  6°  24'  36",  to  find  the  corre 
spending  quantity  in  Table  L. 

7s-  6°  gives  0°  43'  17" .4. 

The  difference  between  0°  43'  17". 4  and  the  next  following  quan- 
tity in  the  table  is  1'  7".3. 

60'  :  1'  7".3  :  :  24'  36"  :  27".6.* 

*  The  student  can  work  the  proportion,  either  by  the  common  method,  or  by  lo- 
gistical logarithms,  as  he  may  prefer.  In  working  this  and  all  similar  proportions 
by  the  arithmetical  method,  the  seconds  of  the  argument  may  be  converted  into 
ihc  equivalent  decimal  part  of  a  minute  by  means  of  Table  XVII,  (using  the  sec- 
onds  as  if  they  were  minutes.)  It  will  be  sufficient  to  take  the  fraction  to  the 
nearest  tenth 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.         323 

From  0°  43'  IT1  A 
Take  27 .6 


0  42   49  .8 

2.  Given  the  argument  2s-  18°  41'  20",  to  find  the  corresponding 
quantity  in  Table  XXV. 

2s-  18°  gives  1°  52'  32". 5. 

The  difference  between  1°  52'  32". 5  and  the  next  following 
quantity  in  the  table  is  21  ".8. 

60' :  21".8  :  :  41'  20"  :  15".0. 

To     1°  52'  32".5 
Add  15  .0 


1    52  47  .5 

3.  Given  the  argument  9*  2°  13'  33",  to  find  the  correspond- 
ing quantity  in  Table  XII. 

9s-  2°  gives  29.8s. 

The  arithmetical  sum  of  29.8s.  and  the  next  following  quantity 
in  the  table  is  30.4s. 

60'  :  30.4s.  :  :  13°  33'  :  6.9s. 

From     29.8s. 
Take       6.9 

22.9s. 
Ans.  —  22.9s 

4.  Given  the  argument  5*  8°  14'  52",  to  find  the  corresponding 
quantity  in  Table  LII.  Ans.  12y  36".0. 

5.  Given  the  argument  II8-  11°  23'  10",  to  find  the  correspond- 
ing quantity  in  Table  LVI.  Ans.  11;  48'  .0. 

6.  Given  the  argument  08<  26°  20',  to  find  the  corresponding 
quantity  in  Table  XII.  Ans.  —  41B.0. 

Case  2.  When  the  argument  changes  in  the  table  by  more  or 
less  than  1°;  or  when  it  is  given  in  lower  denominations  than 
signs. 

Take  out  of  the  table  the  quantity  answering  to  the  number  in 
the  column  of  arguments  next  less  than  the  given  argument.  Talie 
the  difference  between  this  quantity  and  the  next  following  one, 
and  also  the  difference  of  the  consecutive  values  of  the  argument 
inserted  in  the  table,  and  say,  difference  ,of  arguments  :  difference 
of  quantities  :  :  excess  of  the  given  argument  over  the  value  next 
less  in  the  table  :  a  fourth  term.  This  fourth  term  applied  to  the 
quantity  first  taken  out,  according  to  the  rule  given  in  the  prece- 
ding case,  will  give  the  quantity  sought. 

Note.  In  some  of  the  tables  the  columns  entitled  Diff.  are  made 
up  of  the  differences  answering  to  a  difference  of  10'  in  the  argu- 
ment. In  obtaining  quantities  from  these  tables,  it  will  be  found 
more  convenient  to  take  for  the  first  and  second  terms  of  the  pro 


324  ASTRONOMICAL  PROBLEMS. 

portion,  respectively,  10',  and  the  difference  furnished  by  the  table 
and  work  the  proportion  by  the  arithmetical  method.    (See  note  at 
bottom  of  page  268.) 

Exam.  1.  Given  the  argument  0s-  24°  42'  15",  to  find  the  cor- 
responding quantity  in  Table  LI. 

0s-  24°  30' gives  9°  47'  14".3. 

The  difference  between  9°  47'  14".3  and  the  next  following 
quantity  =  3  x  63".0  =  189".0.  The  argument  changes  by  30'. 
And  the  excess  of  0s-  24°  42'  15"  over  0s-  24°  30',  is  12  15".  Thus, 

30'  :  189".0  :  :  12' 15"  :  77".2. 

But  the  correction  may  be  found  more  readily  by  the  following 
proportion : 

10'  :  63".0  :  :  12'.25  :  77".2. 

To     9°  47'  14".3 
Add  77  .2 


9   48  31  .5 

2.  Given  the  argument  1°  12',  to  find  the  corresponding  quan- 
tity in  Table  VIII. 

1°  10'  gives  23'  13", 
and  5'  :  33"  :  :  2'  :  13"  the  correction. 

From         23'  13" 
Take  13 

23    0 

3.  Given  the  argument  6s-  6°  7'  23",  to  find  the  corresponding 
quantity  in  Table  LV.  Ans.  90°  20'  53".5. 

4.  Given  the  argument  49°  27',  to  find  the  corresponding  quan 
tity  in  Table  LXIV.  Ans.  11;  19".8. 

Case  3.   When  the  argument  is  given  in  the  table  in  hundredth, 
thousandth,  or  ten  thousandth  parts  of  a  circle. 

The  required  quantity  can  be  found  in  this  case  by  the  same 
rule  as  in  the  preceding ;  but  it  can  be  had  more  expeditiously  by 
observing  the  following  rules.  If  the  argument  varies  by  10,  mul- 
tiply the  difference  of  the  quantities  between  which  the  required 
quantity  lies  by  the  excess  of  the  given  argument  over  the  next  less 
value  in  the  table,  and  remove  the  decimal  point  one  figure  to  the 
left ;  the  result  will  be  the  correction  to  be  applied  to  the  quantity 
taken  out  of  the  table.  The  same  rule  will  apply  in  taking  quan- 
tities from  tables  in  which  the  differences  answering  to  a  change  of 
10  in  the  argument  are  given,  although  the  argument  should  actu- 
ally change  by  50  or  100.  If  the  argument  changes  by  100,  mul 
tiply  as  above,  and  remove  the  decimal  point  two  figures  to  the  left. 
When  the  common  difference  of  the  arguments  is  5,  proceed  as  if 
it  were  10,  and  double  the  result.  In  like  manner,  when  the  com- 
mon difference  is  50,  proceed  as  if  it  were  100,  and  double  the 
result. 


TO  TAKE  OUT  A  QUANTITY  FROM  A  TABLE.          325 

Exam.  1.  Given  the  argument  973,  to  find  the  corresponding 
quantity  in  Table  XLV  column  headed  13. 

970  gives  23". 5. 
The  difference  is  1". 2,  and  the  excess  3. 

I". 2  From    23".5 

3  Take         .4 


Corr.     .36  23  .1 

2.  Given  the  argument  4834,  to  find  the  corresponding  quantity 
in  Table  XLII,  column  headed  5. 

4800  gives  2'  3';.7. 

The  difference  is  6". 8,  and  the  excess  34. 
6".8 
34 

From   2'3".7 
2.312        .        .         .     Take         2  .3 

2    1  .4 

3.  Given  the  argument  5444,  to  find  the  corresponding  quan- 
tity in  Table  XLI.  Ans.  15'  37/;.7. 

4.  Given  the  argument  4225,  to  find  the  corresponding  quan- 
tity in  Table  XLIII,  column  headed  8.  Ans.  0'  47".2. 

Case  4.  When  the  table  is  one  of  double  entry,  or  quantities  are 
taken  from  it  by  means  of  two  arguments. 

Take  out  of  the  table  the  quantity  answering  to  the  values  of 
the  arguments  of  the  table  next  less  than  the  given  values  ;  and 
find  the  respective  corrections  to  be  applied  to  it,  due  to  the  ex- 
cess of  the  given  value  of  each  argument  over  the  next  less  value 
in  the  table,  by  the  general  rule  in  the  preceding  case.  These 
corrections  are  to  be  added  to  the  quantity  taken  out,  or  subtracted 
from  it,  according  as  the  quantities  increase  or  decrease  with  the 
arguments. 

Note  1.  If  the  tenths  of  seconds  be  omitted,  the  corrections 
above  mentioned  can  be  estimated  without  the  trouble  of  stating  a 
proportion,  or  performing  multiplications. 

Note  2.  The  rule  above  given  may,  in  some  rare  instances,  give 
a  result  differing  a  few  tenths  of  a  second  from  the  truth.  The 
following  rule  will  furnish  more  exact  results.  Find  the  quanti- 
ties corresponding,  respectively,  to  the  value  of  the  argument  at 
the  top  next  less  than  its  given  value  and  the  other  given  argu- 
ment, and  to  the  value  next  greater  and  the  other  given  argument. 
Take  the  difference  of  the  quantities  found,  and  also  the  difference 
of  the  corresponding  arguments  at  top,  and  say,  difference  of  argu- 
ments :  difference  of  quantities  :  :  excess  of  given  value  of  the 
argument  at  the  top  over  its  next  less  value  in  the  table  :  a  fourth 
term.  This  fourth  term  added  to  the  quantity  first  found,  if  it  is 
less  than  the  other,  but  subtracted  from  it,  if  it  is  greater,  will  give 
the  required  quantity.  The  error  of  the  first  rule  may  be  dimin- 


326  ASTRONOMICAL  PROBLEMS. 

ished  without  any  extra  calculation,  by  attending  to  the  difference 
of  the  quantities  answering  to  the  value  of  the  argument  at  the 
side  next  greater  than  its  given  value  and  the  values  of  the  other 
argument  between  which  its  given  value  lies. 

Exam.  1  .  Given  the  argument  64  at  the  top  and  77  at  the  side 
to  find  the  corresponding  quantity  in  Table  LXXXI. 
50  and  70  give  47".  7. 

The  difference  between  47".7  and  the  next  quantity  below  it 
is  I  "A.  The  excess  of  77  over  70  is  7,  and  the  argument  at  the 
side  changes  by  10. 


-  From   47;/.7 

Corr.  due  excess  7,    .98,  or  1".0.         Take      1.0 

Quantity  corresponding  to  50  and  77,      46  .7 
The  difference  between  47".7  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  3".3.     The  excess  of  64  over  50  is  14, 
and  the  argument  at  the  top  changes  by  50. 
3".3 
14 


From    46;/.7 
Corr.  due  excess  14,   .924  Take      0  .9 

45  .8 

2.  Given  the  argument  223  at  the  top  and  448  at  the  side,  to 
find  the  corresponding  quantity  in  Table  XXX. 

220  and  440  give  16".0. 

The  difference  between  16".0  and  the  quantity  next  below  it 
is  2".2. 

2".2 
8 

2  )  1.76 

From    16/;.0 

Corr.  for  excess  8,         .88,  or  0".9.     Take      0  .9 

Quantity  corresponding  to  220  and  448,  15  .1 
The  difference  between  16".0  and  the  adjacent  quantity  in  the 
next  column  on  the  right  is  0".7. 
0".7 
3 

To     15".l 

Coir  for  excess  3,       £1  Add        .2 


TO  CONVERT  DEGREES,  MINUTES,  ETC.,  INTO  TIME.  327 

3.  Given  the  argument  472  at  the  top  and  786  at  the  side,  to 
ftnd  the  corresponding  quantity  in  Table  XXXI. 

Ans.  9".7. 

4.  Given  the  argument  620  at  the  top  and  367  at  the  side,  to 
find  the  corresponding  quantity  in  Table  LXXXI. 

Ans.  55".2. 

5  Given  the  argument  348  at  the  top  and  932  at  the  side,  to 
find  (by  the  rule  given  in  Note  2)  the  corresponding  quantity  in 
Table  XXXII.  Ans.  15".4. 


PROBLEM  III. 

To  convert  Degrees,  Minutes,  and  Seconds  of  the  Equator  into 
Hours,  Minutes,  fyc.,  of  Time. 

Multiply  the  quantity  by  4,  and  call  the  product  of  the  seconds, 
thirds  ;  of  the  minutes,  seconds  ;  and  of  the  degrees,  minutes. 
Exam.  1.  Convert  83°  II7  52"  into  time. 
83°  11'  52" 
4 


5h. 

2.  Convert  34°  57'  46"  into  time. 

Ans.  2h.  19m.  Slsec.  4"'. 


PROBLEM  IV. 

To  convert  Hours,  Minutes,  and  Seconds  of  Time  into  Degrees, 
Minutes,  and  Seconds  of  the  Equator. 

Reduce  the  hours  and  minutes  to  minutes  :  divide  by  4,  and 
call  the  quotient  of  the  minutes,  degrees  ;  of  the  seconds,  minutes ; 
and  multiply  the  remainder  by  15,  for  the  seconds. 

Exam.  1.  Convert  7h.  9m.  34sec.  into  degrees,  &c. 

7h.  gm.  34.. 

60 


4  )  429  34 


107°  23'  30" 
Convert  1  Ih.  24m.  45s.  into  degrees,  &c. 

Ans.  171°  11' 15" 


328  ASTRONOMICAL  PROBLEMS. 


PROBLEM  V. 

The  Longitudes  of  two  Places,  and  the  Time  at  one  of  them 
being  given,  to  find  the  corresponding  Time  at  the  other. 

When  the  given  time  is  in  the  morning,  change  it  to  astronomi- 
cal time,  by  adding  1 2  hours,  and  diminishing  the  number  of  the 
day  by  a  unit.  When  the  given  time  is  in  the  evening,  it  is  al- 
ready in  astronomical  time. 

Find  the  difference  of  longitude  of  the  two  places,  by  taking  the 
numerical  difference  of  their  longitudes,  when  these  are  of  the 
same  name,  that  is,  both  east  or  both  west ;  and  the  sum,  when 
they  are  of  different  names,  that  is,  one  west  and  the  other  east. 
When  one  of  the  places  is  Greenwich,  the  longitude  of  the  other 
is  the  difference  of  longitude. 

Then,  if  the  place  at  which  the  time  is  required  is  to  the  east 
of  the  place  at  which  the  time  is  given,  add  the  difference  of  longi- 
tude, in  time,  to  the  given  time  ;  but,  if  it  is  to  the  west,  subtract 
the  difference  of  longitude  from  the  given  time.  The  sum  or  re- 
mainder will  be  the  required  time. 

Note.  The  longitudes  used  in  the  following  examples,  are  given 
in  Table  I. 

Exam.  1.    When  it  is  October  25th,  3h.  13m.  22sec.  A.  M.  at 
Greenwich,  what  is  the  time  as  reckoned  at  New  York? 
Time  at  Greenwich,  October,  24d>  15h>  13m-  22s* 
Diff.  of  Long.         ...  4     56      4 

Time  at  New  York        .         .  24    10     17     18P.M. 

2.  When  it  is  June  9th,  5h.  25m.  lOsec.  P.  M.  at  Washington, 
what  is  the  corresponding  time  at  Greenwich  ? 

Time  at  Washington,  June,         9d-  5h-  25m-  10s- 
Diff.  of  Long.         .         .         .          586 

Time  at  Greenwich        .         .     9  10    33      16P.M. 

3.  When  it  is  January  15th,  2h.  44m.  23sec.  P.  M.  at  Paris 
what  is  the  time  at  Philadelphia  ? 

Longitude  of  Paris         .         .         Oh-  9m-  218 .6    E. 
Do.        of  Philadelphia,     .         5    0     39  .6   W. 

5  10       1.2 

Time  at  Paris,  January,         .      15d-  2h-  44m-  23s- 
Diff.  of  Long.        .         .         .  5     10       1 

Time  at  Philadelphia,  .     14  21     34     22 

Or  January  15th,  9h.  34m.  22sec.  A.  M. 

4.  When  it  is  March  31st,  8h.  4m.  21sec.  P.  M.  at  New  Haven, 
what  is  the  corresponding  time  at  Berlin  ? 

Ans.  April  1st,  Ih.  49m.  43sec.  A.  M. 


TO  CONVERT  APPARENT  INTO  MEAN  TIME.  329 

5.  When  it  is  August  10th,  lOh.  32m.  14sec.  A.  M.  at  Boston, 
what  is  the  time  at  New  Orleans  ? 

Ans.  Aug.  10th,  9h.  16m.  4sec.  A.  M. 

6.  When  it  is  noon  of  the  23d  of  December  at  Greenwich,  what 
is  the  time  at  New  York  ? 

Ans.  Dec.  23d,  7h.  3m.  55sec.  A.  M. 


PROBLEM  VI. 

The  Apparent  Time  being  given,  to  find  the  corresponding  Mean 
Time  ;  or  the  Mean  Time  being  given  to  find  the  Apparent. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian  by  the  last  problem.  Then  find  by  the 
tables  the  sun's  mean  longitude  corresponding  to  this  time.  Thus, 
from  Table  XVIII  take  out  the  longitude  answering  to  the  given 
year,  and  from  Tables  XIX,  XX,  and  XXI,  take  out  the  motions 
in  longitude  for  the  given  month,  days,  hours,  and  minutes,  neg- 
lecting the  seconds.  The  sum  of  the  quantities  taken  from  the 
tables,  rejecting  12  signs,  when  it  exceeds  that  quantity,  will  be 
the  sun's  mean  longitude  for  the  given  time. 

With  the  sun's  mean  longitude  thus  found,  take  the  Equation 
of  Time  from  Table  XII.  Then,  when  Apparent  Time  is  given 
to  find  the  Mean,  apply  the  equation  with  the  sign  it  has  in  the 
table  ;  but  when  Mean  Time  is  given  to  find  the  Apparent,  apply 
it  with  the  contrary  sign  ;  the  result  will  be  the  Mean  or  Apparent 
Time  required. 

This  rule  will  be  sufficiently  exact  for  ordinary  purposes,  for 
several  years  before  and  after  the  year  1840.  When  the  given 
date  is  a  number  of  years  distant  from  this  epoch,  take  also  with 
the  sun's  mean  longitude  the  Secular  Variation  of  the  Equation  of 
Time  from  Table  XIII,  and  find  by  simple  proportion  the  variation 
hi  the  interval  between  the  given  year  and  1840.  The  result,  ap- 
plied to  the  equation  of  time  taken  from  Table  XII,  according  to 
its  sign,  if  the  given  time  is  subsequent  to  the  year  1840,  but  with 
the  opposite  sign  if  it  is  prior  to  1840,  will  give  the  equation  of 
time  at  the  given  date,  which  apply  to  the  given  time  as  above 
directed. 

Note  1.  When  the  exact  mean  or  apparent  time  to  within  a 
small  fraction  of  a  second  is  demanded,  take  the  numbers  in  the 
columns  entitled  I,  II,  III,  IV,  V,  N,  in  Tables,  XVIII,  XIX, 
XX,  answering  respectively  to  the  year,  month,  days,  and  hours, 
of  the  given  time.  With  the  respective  sums  of  the  numbers 
taken  from  each  column,  as  arguments,  enter  Table  XIV,  and 
take  out  the  corresponding  quantities.  These  quantities  added  to 
the  equation  of  time  as  given  by  Tables  XII  and  XIII,  and  uie 


330  ASTRONOMICAL  PROBLEMS. 

constant  3.0s.  subtracted,  will  give  the  true  Equation  of  Time,  if 
the  given  time  is  Mean  Time.  When  Apparent  Time  is  given,  it 
will  be  farther  necessary  to  correct  the  equation  of  time  as  gives 
by  the  tables,  by  stating  the  proportion,  24  hours  :  change  of 
equation  for  1°  of  longitude  :  :  equation  of  time  :  correction. 

Note  2.  The  Equation  of  Time  is  given  in  the  Nautical  Alma- 
nac for  each  day  of  the  year,  at  apparent,  and  also  at  mean  noon, 
on  the  meridian  of  Greenwich,  and  can  easily  be  found  for  any 
intermediate  time  by  a  proportion.  Directions  for  applying  it  to  the 
given  time  are  placed  at  the  head  of  the  column.  The  Equation 
is  given  on  the  first  and  second  pages  of  each  month. 

Exam.  1.  On  the  16th  of  July,  1840,  when  it  is  9h.  35m.  22s. 
P  M.,  mean  time  at  New  York,  what  is  the  apparent  time  at  the 
same  place  ? 

Time  at  New  York,  July,  1840,      16d-  9h-  35m-  22"- 
Diff.  of  Long.        ...  4    56      4 

Time  at  Greenwich,  July,  1840,     16  14    31     26 

M.  Long. 
1840       .         .  .         9»- 10°  12' 49' 


July 
16d. 
14h. 
31m. 


5    29   23  16 

14  47     5 

34  30 

1    16 


M.  Long.       .  .         .         3    24   58  56 

The  equation  of  time  in  Table  XII,  corresponding  to  3**  24"  58 
56",  is  +  5m-  44"- 

Mean  Time  at  New  York,  July,  1840,  16d-  9h-  35m-  22s- 
Equation  of  time,  sign  changed,         .  — 5     44 

Apparent  Time,       .        .        .         .  16    9    29     38P.M. 

2.  On  the  9th  of  May,  1842,  when  it  is  4h.  15m.  21  sec.  A.  M. 

apparent  time  at  New  York,  what  is  the  mean  time  at  the  same 

place,  and  also  at  Greenwich  ? 

Time  at  New  York,  May,  1842,     8d-  16h-  15m-  21fc 
Diff.  of  Long.         ...  4     56      4 

Time  at  Greenwich,  .        .        8   21     11    25 

M.  Long. 

1842      .         .         9"-  10°  43'  18" 

May       .         .         3  28    16  40 

8d.  6   53  58 

2lh.       .         .  51  45 

llin.  27 


M.  Long        .         1     16  46     8.  Equa.  of  time,— 3m.  45s, 


TO  CONVERT  APPARENT  INTO  MEAN  TIME. 


331 


Apparent  Time  at  Greenwich,  May,  1842, 
Equation  of  Time,         .... 

Mean  Time  at  Greenwich,     . 

Diff.  of  Long 


21h. 


-3     45 


8     21      7    40 
4    56       4 

8     16    11     36 


Mean  Time  at  New  York,     . 
Or,  May  9th,  4h.  llm.  36s.  A.  M. 

3.  On  the  3d  of  February,  1855,  when  it  is  2h.  43m  36s.  appa- 
rent time  at  Greenwich,  what  is  the  exact  mean  time  at  the  same 
place  ? 

Appar.  Time  at  Greenwich,  Feb.,  1855,  3d.  2h.  43m.  36s 


M.  Long. 

I. 

n. 

ra. 

rv. 

V. 

N. 

1855 

9'  10°  34'  30" 

433 

279 

806 

889 

866 

863 

Feb. 

1   0  33  18 

47 

85 

138 

45 

7 

5 

3d. 

1  58  17 

68 

5 

9 

3 

0 

0 

2h. 

4  56 

3 

43m. 

1  46 

10  13  12  47 

551 

369 

953 

937 

873 

868 

Appar.  Time  at  Greenwich,  Feb.,  1855,  3d-  2h-  43m-  361- 
Equation  of  time  by  Table  XII,  .  +14       8.6 

lOOyrs.  :  13s.  (Sec.  Var.,  Table  XIII) 

::  15yrs.:  1.9s.          ...  —1.9 


Approx.  Mean  Time  at  Greenwich,      .   3    2    57    42.7 
24h.  :  6s.  (change  of  equa.  for  1°  of 

long.)::  14m.:  O.ls.         .         .  +0.1 

II.  III.       .....  0.8 

II.  IV  ......  0.4 

II.  V  ......  1.0 

1  .......  0.3 

N  .......  0.1 

Constant.                     .                 .  —  3.0 


Mean  Time  at  Greenwich,  3    2    57    42.4 

4.  On  the  18th  of  November,  1841,  when  it  is  2h.  12m.  26sec. 
A.  M.  mean  time  at  Greenwich,  what  is  the  apparent  time  at 
Philadelphia?  Ans.  Nov.  17th,  9h.  26m.  28s.  P.  M. 

5.  On  the  2d  of  February,  1839,  when  it  is  6h.  32m.  35sec. 
P.  M.,  apparent  time  at  New  Haven,  what  is  the  mean  time  at  tho 
same  place  ?  Ans.  6h.  46m.  39s.  P.  M. 

6.  On  the  23d  of  September,  1850,  when  it  is  9h.  10m.  12sec. 
mean  time  at  Boston,  what  is  the  exact  apparent  time  at  the  same 
place?  Ans.  9h.  18m.  1.0s. 


332  ASTRONOMICAL  PROBLEMS. 


PROBLEM  VII. 

To  correct  the  Observed  Altitude  of  a  Heavenly  Body  for  Re  • 

fraction. 

With  the  given  altitude  take  the  corresponding  refraction  from 
Table  VIII.  Subtract  the  refraction  from  the  given  altitude,  and 
the  result  will  be  the  true  altitude  of  the  body  at  the  given  station. 

This  rule  will  give  exact  results  if  the  barometer  stands  at  30 
inches,  and  Fahrenheit's  thermometer  at  50°,  and  results  suffi- 
ciently exact  for  ordinary  purposes  in  any  state  of  the  atmosphere. 
When  there  is  occasion  for  greater  precision,  take  from  Table  IX 
the  corrections  for  +  1  inch  in  the  height  of  the  barometer,  and 
—  1°  in  the  height  of  Fahrenheit's  thermometer,  and  compute  the 
corrections  for  the  difference  between  the  observed  height  of  the 
barometer  and  30in.  and  for  the  difference  between  the  observed 
height  of  the  thermometer  and  50°.  Add  these  to  the  mean  re- 
fraction taken  from  Table  VIII,  if  the  barometer  stands  higher 
than  30in.  and  the  thermometer  lower  than  50°  ;  but  in  the  oppo- 
site case  subtract  them,  and  the  result  will  be  the  true  refraction, 
which  subtract  from  the  observed  altitude. 

Exam.  1.  The  observed  altitude *of  the  sun  being  32°  10'  25", 
what  is  its  true  altitude  at  the  place  of  observation  ? 

Observed  alt.         .         .         .         32°  10'  25" 
Refraction  (Table  VIII)         .  — 1  32 

True  alt.  at  the  station,  .         32°    853 

2.  The  observed  altitude  of  Sirius  being  20°  42'  11",  the  ba- 
rometer 29.5  inches,  and  the  thermometer  of  Fahrenheit  70°, 
required  the  true  altitude  at  the  place  of  observation.  The  differ- 
ence between  29.5  inches  and  30  inches  is  0.5  inches,  and  the 
difference  between  70°  and  50°  is  20°. 
Obs.  alt.  20°42'11".0 


Refrac. (Table VIII),  2'  33".0;  Bar.+lm.,5".12;  ther.-l°.0".310 
Corr.for— 0.5  in.,  bar.  —2  .6  .5  20 

Corr.for+20°,ther.      -6  .2  • 

2.560  6.20 

True  refrac.  2  24  .2 


True  alt.  20  39  46  .8 

3.  The  observed  altitude  of  the  moon  on  the  llth  of  April,  1838, 
being  14°  17;  20",  required  the  true  altitude  at  the  place  of  obser- 
vation. Ans.  14°  13'  35". 

4.  Let  the  observed  altitude  of  Aldebaran  be  48°  35'  52",  the 
barometer  at  the  same  time  standing  at  30  7  inches,  and  the  ther- 
mometer at  42°,  required  the  true  altitude.  Ans.  48°  34'  58".8. 


TO  DEDUCE  THE  TRUE  FROM  THE  APPARENT  ALTITUDE.   333 


PROBLEM  VIII. 

The  Apparent  Altitude  of  a  Heavenly  Body  being  given,  to  find 
its  True  Altitude. 

Correct  the  observed  altitude  for  refraction  by  the  foregoing 
problem.  Then, 

1 .  If  the  sun  is  the  body  whose  altitude  is  taken,  find  its  paral- 
lax in  altitude  by  Table  X,  and  add  it  to  the  observed  altitude  cor- 
rected for  refraction.     The  result  will  be  the  true  altitude  sought, 

2.  If  it  is  the  altitude  of  the  moon  that  is  taken,  and  the  hori- 
zontal parallax  at  the  time  of  the  observation  is  known,  find  the 
parallax  in  altitude  by  the  following  formula : 

log.  sin  (par.  in  alt.)  =  log.  sin  (hor.par.)  -Hog.  cos  (app.alt.)  — 10 ; 

and  add  it,  as  before,  to  the  apparent  altitude  corrected  for  refrac- 
tion. 

3.  If  one  of  the  planets  is  the  body  observed,  the  following  for- 
mula will  serve  for  the  determination  of  the  parallax  in  altitude 
when  the  horizontal  parallax  is  known : 

log.  (par.  in  alt.)  =  log.  (hor.  par.)  +  log.  cos  (appar.  alt) — 10. 

Note  1.  The  equatorial  horizontal  parallax  of  the  moon  at  any 
given  time  may  be  obtained  from  the  tables  appended  to  the  work. 
(See  Problem  XIV.)  But  it  can  be  had  much  more  readily  from 
the  Nautical  Almanac.  The  equatorial  horizontal  parallax  being 
known,  the  horizontal  parallax  at  any  given  latitude  may  be  ob- 
tained by  subtracting  the  Reduction  of  Parallax,  to  be  found  in 
Table  LXIV.  The  horizontal  parallax  of  any  planet,  the  altitude 
of  which  is  measured,  may  also  be  derived  from  the  Nautical  Al- 
manac. 

Note  2.  The  fixed  stars  have  no  sensible  parallax,  and  thus  the 
observed  altitude  of  a  star,  corrected  for  refraction,  will  be  its  true 
altitude  at  the  centre  of  the  earth  as  well  as  at  the  station  of  the 
observer. 

Note  3.  If  the  true  altitude  of  a  heavenly  body  is  given,  and  it 
is  required  to  find  the  apparent,  the  rules  for  finding  the  parallax 
in  altitude  and  the  refraction  are  the  same  as  when  the  apparent 
altitude  is  given  ;  the  true  altitude  being  used  in  place  of  the  ap- 
parent. But  these  corrections  are  to  be  applied  with  the  opposite 
signs  from  those  used  in  the  determination  of  the  true  altitude  from 
the  apparent ;  that  is,  the  parallax  is  to  be  subtracted,  and  the  re- 
fraction added.  It  will  also  be  more  accurate  to  make  use  of 
equa.  (a),  p.  422,  in  the  case  of  the  moon. 

Exam.  1.  The  observed  altitude  of  the  sun  on  the  1st  of  May 
1837,  being  26°  40'  20",  what  is  its  true  altitude? 


334 


ASTRONOMICAL  PROBLEMS. 


Obs.  alt. 
Refraction     . 

True  alt.  at  the  station, 
Parallax  in  alt.  (Table  X), 


26'  40'  20" 
—  1  56 

26    38  24 

+  8 


True  altitude         .         .         .         26    38  32 

2.  Let  the  apparent  altitude  of  the  moon  at  New  York  on  the 
17th  of  March,  1837,  8h.  P.  M.,  be  66°  10'  44" ;  the  barometer 
30.4in.  and  the  thermometer  62°  ;  required  the  true  altitude. 
Appar.  alt.  .         .         66°  10'  44/; 

Meanrefrac.         .  0  25.7 

Corr.  for  +  0.4in.,  bar.  -j-  0.3 

Corr.  for  +  12°,  ther.  —0.6 


True  refrac. 


0  25.4 


True  alt.  at  N.  York,    66  10  18.6 
Equa.  par.  by  N.  Almanac,  54'  13" 
Reduc.  for  lat.  40°,  4 


Hor.  par.  at  New  York,       54     9 
Par.  in  alt. 


21  52 


logarithms 
cos.  9.60637 


sin.  8.19731 
sin.  7.80368 


True  altitude     .         .  66  32  11 

3.  On  the  18th  of  February,  1837,  the  true  meridian  altitude  of 
the  planet  Jupiter  at  Greenwich  was  56°  54'  57",  what  was  its 
apparent  altitude  at  the  time  of  the  meridian  passage,  the  horizontal 
parallax  being  taken  at  1".9,  as  given  by  the  Nautical  Almanac  ? 

True  alt.  .         .         56°  54'  57''     .     cos.  9.7371 

Hor.  par.  1".9  .         .         .         .         .  log.  0.2787 


Par.  in  alt. 
Refraction 


—1.0 
+  37.9 


log.  0.0158 


Appar.  alt.         .         .         56    55  34 

4.  What  will  be  the  true  altitude  of  the  sun  on  the  22d  of  Sep- 
tember, 1840,  at  the  time  its  apparent  altitude  is  39°  17'  50"  ? 

Ans.  39°  16'  46". 

5.  Given  29°  33'  30"  the  apparent  altitude  of  the  moon  at  Phil 
adelphia  on  the  15th  of  June,  1837,  at  9h.  30m.  P.  M.,  and  58'  33' 
the  equatorial  parallax  of  the  moon  at  the  same  time,  to  find  the 
true  altitude.  Ans.  30°  22'  41". 

6.  Given  15°  24'  23"  the  true  altitude  of  Venus,  and  8"  its  hori- 
zontal parallax,  to  find  the  apparent  altitude    Ans.  15°  27'  41' . 


TO  FIND  THE  Sim's  LONGITUDE,  ETC  ,  FROM  TABLES'.         335 


PROBLEM  IX. 

Tojind  the  Surfs  Longitude,  Hourly  Motion,  and  Semi-diameter, 
for  a  given  time,  from  the  Tables. 

For  the  Longitude. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian  by  Problem  V ;  and  when  it  is  apparent 
time,  convert  it  into  mean  time  by  the  last  problem. 

With  the  mean  time  at  Greenwich,  take  from  Tables  XVIII, 
XIX,  XX,  and  XXI,  the  quantities  corresponding  to  the  year, 
month,  day,  hour,  minute,  and  second,  (omitting  those  in  the  last 
two  columns,)  and  place  them  in  separate  columns  headed  as  in 
Table  XVIII,  and  take  their  sums.*  The  sum  in  the  column  enti- 
tled M.  Long,  will  be  the  tabular  mean  longitude  of  the  sun ;  the 
sum  in  the  column  entitled  Long.  Perigee  will  be  the  tabular  lon- 
gitude of  the  sun's  perigee  ;  and  the  sums  in  the  columns  I,  II, 
III,  IV,  V,  N,  will  be  the  arguments  for  the  small  equations  of  the 
sun's  longitude,  including  the  equation  of  the  equinoxes  in  longi- 
tude. 

Subtract  the  longitude  of  the  perigee  from  the  sun's  mean  longi- 
tude, adding  12  signs  when  necessary  to  render  the  subtraction 
possible ;  the  remainder  will  be  the  sun's  mean  anomaly.  With 
the  mean  anomaly  take  the  equation  of  the  sun's  centre  from  Ta- 
ble XXV,  and  correct  it  by  estimation  for  the  proportional  part  of 
the  secular  variation  in  the  interval  between  the  given  year  and 
1840;  also  with  the  arguments  I,  II,  III,  IV,  V,  take  the  corre- 
sponding equations  from  Tables  XXVIII,  XXX,  XXXI,  and 
XXXII.  The  equation  of  the  centre  and  the  four  other  equations, 
together  with  the  constant  3",  added  to  the  mean  longitude,  will 
give  the  sun's  True  Longitude,  reckoned  from  the  Mean  Equinox. 

With  the  argument  N  take  the  equation  of  the  equinoxes  or  Lu- 
nar Nutation  in  Longitude  from  Table  XXVII.  Also  take  the  So- 
lar Nutation  in  longitude,  answering  to  the  given  date,  from  the 
same  table.  Apply  these  equations  according  to  their  signs  to  the 
true  longitude  from  the  mean  equinox,  already  found ;  the  result 
will  be  the  True  Longitude  from  the  Apparent  Equinox. 

For  the  Semi-diameter  and  Hourly  Motion. 

With  the  sun's  mean  anomaly,  take  the  hourly  motion  and  semi- 
diameter  from  Tables  XXIII  and  XXIV. 

*  In  adding  quantities  that  are  expressed  in  signs,  degrees,  &c.,  reject  12  or  24 
signs  whenever  the  sum  exceeds  either  of  these  quantities.  In  adding  arguments 
expressed  in  100  or  1000,  &c.  parts  of  the  circle,  when  they  consist  of  two  figures 
reject  the  hundreds  from  the  sum ;  when  of  three  figures,  the  thousands ;  and 
when  of  four  figures,  the  ten  thousands. 


336 


ASTRONOMICAL  PROBLEMS. 


Notes. 

1 .  If  the  tenths  of  seconds  be  omitted  in  taking  the  equations 
from  the  tables  of  double  entry,  the  error  cannot  exceed  2" ;  in 
case  the  precaution  is  taken  to  add  a  unit,  whenever  the  tenths  ex- 
ceed .5. 

2.  The  longitude  of  the  sun,  obtained  by  the  foregoing  rule, 
may  differ  about  3"  from  the  same  as  derived  from  the  most  accu- 
rate solar  tables  now  in  use.     When  there  is  occasion  for  greater 
precision,  take  from  Tables  XVIII,  XIX,  and  XX,  the  quantities 
in  the  columns  entitled  VI  and  VII,  along  with  those  in  the  other 
columns.    With  the  sums  in  these  columns,  and  those  in  the  col- 
umns I,  II,  as  arguments,  take  the  corresponding  equations  from 
Tables  XXIX  and  XXXIII.     Also  with  the  sun's  mean  anomaly 
take  the  equation  for  the  variable  part  of  the  aberration  from  Ta- 
ble XXXIV.     Add  these  three  equations  along  with  the  others  to 
the  mean  longitude,  and  omit  the  addition  of  the  constant  3".    The 
result  will  be  exact  to  within  a  fraction  of  a  second. 

Exam.  1.  Required  the  sun's  longitude,  hourly  motion,  and  se 
mi-diameter,  on  the  25th  October,  1837,  at  llh.  27m.  38s.  A.  M 
mean  time  at  New  York. 

Mean  time  at  N.  York,  Oct.  1837,  24d-  23h-  27m-  38s- 
Diff.  of  Long 4     56        4 


Mean  time  at  Greenwich,    . 


25     4     23      42 


1837    . 

October 
25d.  . 
4h.  . 
23m.  . 
42s.  . 


Eq.  Sun's  Cent. 

II.  III. 
II.  IV. 
II.  V. 

Const.  . 


Lunar  Nutation 
Solar  Nutation 

Sun'strue  long, 


M.  Long. 


Long.  Perigee.  I. 


9  10  55  47.2 

8  29     4  54.1 

23  39  19.9 

9  51.4 

56.7 

1.7 


7    3  50  51.0 

11  28  12  43.5 

2.5 

9.0 

7.7 

19.3 

3.0 


7     2 


4  16.0 

—  6.3 

—  1.2 


7     2    4    8.5 


9  10    8     5'816280 


46  250  748  215  397 


4810    66 

j     6      0 


9  10     8  55882    94 
7     3  50  51 


III. 


549  321 


107 


872 


IV. 


35 


753 


V. 


348  895 


63 


416 


N. 


40 


939 


9  23  41  56  Mean  Anomaly. 
Sun's  Hourly  Motion,      .     .  2'  29".7 
Sun's  Semi-diameter       .     16'  17".2 


2.  Required  the  sun's  longitude,  hourly  motion,  and  eemi-diam 
eter,  on  the  15th  of  July,  1837,  at  8h.  20m.  40s.  P.  M  mean  time 
at  Greenwich. 


TO  FIND  THE  APPARENT  OBLIQUITY  OF  THE  ECLIPTIC.        337 


1 

1837 
July 
15d. 
8h. 
20m. 
40s.  . 

Eq.  Sun's  Cent. 

II.  III. 
II.    IV. 
II.     V. 
I.   VI. 
II.  VII. 
Aber.    . 

Lunar  Nutation 
Solar  Nutation 

Sun's  true  long. 

M.  Long. 

Long.  Peri. 

i.  !  11.  in  iv.  v. 

N.  VI.  VII. 

S        O           1             /' 

9  10  55  47.2 
5  28  24    7.8 
13  47  56.6 
19  42.8 
49.3 
1.6 

g        0         ; 

9  10    8     5 
31 
2 

816280549321348 
129496806263  41 
473  38!  62|  20|    3 

nil     | 

895787   600 
27  569     17 
2508       2 

11 

9  10    8  38 
3  23  28  25 

429l815l418  604392  9241875'  619 

3  23  28  25.3 
11  29  33  10.3 
10.7 
6.6 
5.0 
7.7 
1.8 
0.2 
0.6 

6  13  19  47  Mean  Anomaly. 
Sun's  Hourly  Motion,     .                 2'  23'  1 
Sun's  Semi-diameter,                      15'  45".4 

3  23    2    8.2 
—  7.8 
+  0.8 

3  23    2     1.2 

3.  Required  the  sun's  longitude,  hourly  motion,  and  semi-diam- 
eter, on  the  10th  of  June,  1838,  at  9h.  45m.  26s.  A.  M.  mean  time 
at  Philadelphia,  (omitting  the  three  smallest  equations  of  longi- 
tude.) 

Ans.   Sun's  longitude,  2s- 19°  11'  57"  ;  hourly  motion,  2'  23".3  ; 
semi-diameter,  15'  46". 1. 

4.  Required  the  sun's  longitude,  hourly  motion,  and  semi-diam 
eter,  on  the  1st  of  February,  1837,  at  12h.  30m.  15s.  mean  astro 
nomical  time  at  Greenwich. 

Ans.    Sun's  longitude,  10s-  13°  1'  44".6;  hourly  motion,  2 
32". 1;  semi-diameter,  16'  14/;.7. 


PROBLEM  X. 


To  find,  the  Apparent  Obliquity  of  the  Ecliptic,  for  a  given  time, 
from  the  Tables. 

Take  the  mean  obliquity  for  the  given  year  from  Table  XXII. 
Then  with  the  argument  N,  found  as  in  the  foregoing  problem, 
and  the  given  date,  take  from  Table  XXVII  the  lunar  and  solar 
nutations  of  obliquity.  Apply  these  according  to  their  signs  to  the 
ity,  and  the  result  will  be  the  apparent  obliquity. 


mean 


Exam.  1 .  Required  the  apparent  obliquity  of  the  ecliptic  on 
1 5th  of  March,  1839. 


338  ASTRONOMICAL  PROBLEMS. 

N. 

1839,  .  3 
March,  9 
15d.  .  2 

M.  Obliquity,       23°  27'  36".9 

14          ...  +9  .1 

Solar  Nutation  for  March  15th,  -j-0  .5 


Apparent  Obliquity,     .         .         23  27  46  .5 
2.  Required  the  apparent  obliquity  of  the  ecliptic  on  the  12th 
af  July,  1845.  Ans.  23°  27'  28".2. 

PROBLEM  XL 

Given  the  Surfs  Longitude  and  the  Obliquity  of  the  Ecliptic,  tc 
Jind  his  Right  Ascension  and  Declination* 

Let  w  =  obliquity  of  the  ecliptic  ;  L  =  sun's  longitude  ;  R  = 
sun's  right  ascension ;  and  D  =  sun's  declination ;  then  to  find  R 
and  D,  we  have 

log.  tang  R  =  log.  tang  L  +  log.  cos  w  —  10, 
log.  sin  D  =  log.  sin  L  +  log.  sin  w  —  10. 

The  right  ascension  must  always  be  taken  in  the  same  quadrant 
as  the  longitude.  The  declination  must  be  taken  less  than  90°  ; 
and  it  will  be  north  or  south  according  as  its  trigonometrical  sine 
comes  out  positive  or  negative. 

Note.  The  sun's  right  ascension  and  declination  are  given  in 
the  Nautical  Almanac  for  each  day  in  the  year  at  noon  on  the  me- 
ridian of  Greenwich,  and  may  be  found  at  any  intermediate  time 
by  a  proportion. 

Exam.  1.  Given  the  sun's  longitude  205°  23'  50",  and  the  ob- 
•iquity  of  the  ecliptic  23°  27'  36",  to  find  his  right  ascension  and 
declination. 

L=205°  23'    50;/       .         .         .         tan.   9.67649 
w  =    23     27     36  .         .         cos.   9.96253 


R  =  203     32       5  .         tan.   9.63902 


L  =  205     23  50         .         .  .  sin.    9.63235— 

w=    23     27  36         .         .  .  sin.    9.60000 

D=      9     49  52  S.   .         .  .  sin.    9.23235  — 

2.  The  obliquity  of  the  ecliptic  being  23°  27'  30",  required 


*  The  obliquity  of  the  ecliptic  at  any  given  time  for  which  the  sun's  longitude 
is  known,  is  found  by  the  foregoing  Problem. 


TO  FIND  THE  SUN  S  LONGITUDE  AND  DECLINATION.  339 

the  sun's  right  ascension  and  declination  when  his  longitude  is 
44°  18'  25". 

Aits.  Right  ascension  41°  50'  30",  and  declination  16°8;40"N. 


PROBLEM  XII. 

Given  the  Sun's  Right  Ascension  and  the  Obliquity  of  the  Eclip 
tic,  to  find  his  Longitude  and  Declination. 

Using  the  same  notation  as  in  the  last  problem,  we  have,  to  find 
the  longitude  and  declination, 

log.  tang  L  =  log.  tang  R  +  ar.  co.  log.  cos  w, 
log.  tang  D  =  log.  sin  R  +  log.  tang  w  —  10. 

Exam.  1.  What  is  the  longitude  and  declination  of  the  sun, 
when  his  right  ascension  is  142°  IT  34",  and  the  obliquity  of  the 
ecliptic  23°  27'  40"  ? 

R  =  142°    11;    34"  tan.    9.88979- 

w  =   23     27     40         .         .      ar.  co.  cos.    0.03747 

L  =  139     46     30         .         .         .         tan.     9.92726  — 

R=  142     11     34         .         .         .         sin.     9.78746 
w=   23     27     40         .         .         .         tan.    9.63750 

D=    14     53     55  N.    .         .         .         tan.    9.42496 

2.  Given  the  sun's  right  ascension  310°  25'  11",  and  the  obli- 
quity of  the  ecliptic  23°  27'  35",  to  find  the  longitude  and  declina- 
tion. 

Ans.  Longitude  307°  59'  57",  and  declination  18°  17'  0"  S. 


PROBLEM   XIII. 

The  Sun's  Longitude  and  the  Obliquity  of  the  Ecliptic  being 
given,  to  find  the  Angle  of  Position. 

Let  p  =  angle  of  position  ;  w  =  obliquity  of  the  ecliptic  ;  and 
L  =  sun's  longitude.  Then, 

log.  tangp  =  log.  cos  L  +  log.  tang  u  —  10. 

The  angle  of  position  is  always  less  than  90°.  The  northern 
part  of  the  circle  of  latitude  will  lie  on  the  west  or  east  side  of  the 
northern  part  of  the  circle  of  declination,  according  as  the  sign  of 
the  tangent  of  the  angle  of  position  is  positive  or  negative. 

Exam.  1.  Given  the  sun's  longitude  24°  15'  20",  and  the  obli- 
quity of  the  ecliptic  23°  27'  32",  required  the  angle  of  position. 


340  ASTRONOMICAL  PROBLEMS. 

L  =  24°  15'  20"   .    .    cos,  9.95980 
w  =  23  27  32    .    .    tan.  9.63745 

^=21     35     10         .         .         tan.    9.59731 

The  northern  part  of  the  circle  of  latitude  is  to  the  west  of  the 
circle  of  decimation. 

2.  When  the  sun's  longitude  is  120°  18'  55",  and  the  obliquity 
of  the  ecliptic  23°  27'  30  ',  what  is  the  angle  of  position? 

Ans.  12°  21'  17'*  ;  and  the  northern  part  of  the  circle  of  latitude 
lies  to  the  east  of  the  circle  of  declination. 


PROBLEM  XIV. 

To  find  from  the  Tables,  the  Moon's  Longitude,  Latitude,  Equa- 
torial Parallax,  Semi-diameter,  and  Hourly  Motion  in  Longi- 
tude and  Latitude,  for  a  given  time. 

When  the  given  time  is  not  for  the  meridian  of  Greenwich,  re- 
duce it  to  that  meridian,  and  when  it  is  apparent  time  convert  it 
into  mean  time. 

Take  from  Table  XXXV,  and  the  following  tables,  the  argu- 
ments numbered  1 ,  2,  3,  &c.,  to  20,  for  the  given  year,  and  their 
variations  for  the  given  month,  days,  &c.,  and  find  the  sums  of  the 
numbers  for  the  different  arguments  respectively ;  rejecting  the 
hundred  thousands  and  also  the  units  in  the  first,  the  ten  thousands 
in  the  next  eight,  and  the  thousands  in  the  others.  The  resulting 
quantities  will  be  the'  arguments  for  the  first  twenty  equations  of 
longitude. 

With  the  same  time,  take  from  the  same  tables  the  remaining 
arguments  with  their  variations,  entitled  Evection,  Anomaly,  Va- 
riation, Longitude,  Supplement  of  the  Node,  II,  V,  VI,  VII,  VIII, 
IX,  and  X ;  and  add  the  quantities  in  the  column  for  the  Supple- 
ment of  the  Node. 

For  the  Longitude. 

With  the  first  twenty  arguments  of  longitude,  take  from  Tables 
XLI  to  XLVI,  inclusive,  the  corresponding  equations  ;  and  with 
the  Supplement  of  the  Node  for  another  argument,  take  the  corre- 
sponding equation  from  Table  XLIX.  Place  these  twenty-one 
equations  in  a  single  column,  entitled  Eqs.  of  Long. ;  and  write 
beneath  them  the  constant  55".  Find  the  sum  of  the  whole,  and 
place  it  in  the  column  of  Evection.  Then  the  sum  of  the  quanti 
ties  in  this  column  will  be  the  corrected  argument  of  Evection. 

With  the  corrected  argument  of  Evection,  take  the  Evection 
from  Table  L,  and  add  it  to  the  sum  in  the  column  of  Eqs.  of 
Long.  Place  this  in  the  column  of  Anomaly.  Then  the  sum  of 
the  quantities  in  this  column  will  be  the  corrected  Anomaly. 


TO  FIND  THE  MOGN's  LONGITUDE,   ETC.  34:1 

With  the  corrected  Anomaly,  take  the  Equation  of  the  Centre 
from  Table  LI,  and  add  it  to  the  last  sum  in  the  column  of  Eqs. 
of  Long.  Place  the  resulting  sum  in  the  column  of  Variation. 
Then  the  sum  of  the  quantities  in  this  column  will  be  the  corrected 
argument  of  Variation. 

With  the  corrected  argument  of  Variation,  take  the  variation 
from  Table  LII,  and  add  it  to  the  last  sum  in  the  column  of  Eqs. 
of  Long. ;  the  result  will  be  the  sum  of  the  principal  equations 
of  the  Orbit  Longitude,  amounting  in  all  to  twenty  four,  and  the 
constants  subtracted  for  the  other  equations.  Place  this  sum  in 
the  column  of  Longitude.  Then  the  sum  of  the  quantities  in  this 
column  will  be  the  Orbit  Longitude  of  the  Moon,  reckoned  from 
the  mean  equinox. 

Add  the  orbit  longitude  to  the  supplement  of  the  node,  and  the 
resulting  sum  will  be  the  argument  of  Reduction. 

With  the  argument  of  Reduction,  take  the  Reduction  from  Ta- 
ble LIII,  and  add  it  to  the  Orbit  Longitude.  The  sum  will  be  the 
Longitude  as  reckoned  from  the  mean  equinox.  With  the  Supple- 
ment of  the  Node,  take  the  Nutation  in  Longitude  from  Table 
LIV,  and  apply  it,  according  to  its  sign,  to  the  longitude  from  the 
mean  equinox.  The  result  will  be  the  Moon's  True  Longitude 
from  the  Apparent  Equinox. 

For  the  Latitude. 

The  argument  of  the  Reduction  is  also  the  1  st  argument  of  Lat- 
itude. Place  the  sum  of  the  first  twenty-four  equations  of  Longi- 
tude, taken  to  the  nearest  minute,  in  the  column  of  Arg.  II.  Find 
the  sum  of  the  quantities  in  this  column,  and  it  will  be  the  Arg.  II 
of  Latitude,  corrected.  The  Moon's  true  Longitude  is  the  3d  ar- 
gument, of  Latitude.  The  20th  argument  of  Longitude  is  the  4th 
argument  of  Latitude.  Take  from  Table  LVIII  the  thousandth 
parts  of  the  circle,  answering  to  the  degrees  and  minutes  in  the 
sum  of  the  first  twenty-four  equations  of  longitude,  and  place  it  in 
the  columns  V,  VI,  VII,  VIII,  and  IX  ;  but  not  in  the  column  X. 
Then  the  sums  of  the  quantities  in  columns  V,  VI,  VII,  VIII,  IX, 
and  X,  rejecting  the  thousands,  will  be  the  5th,  6th,  7th,  8th,  9th, 
and  10th  arguments  of  Latitude. 

With  the  Arg.  I  of  Latitude,  take  the  moon's  distance  from  the 
North  Pole  of  the  Ecliptic,  from  Table  LV ;  arid  with  the  remain- 
ing nine  arguments  of  latitude,  take  the  corresponding  equations 
from  Tables  LVI,  LVII,  and  LIX.  The  sum  of  these  quantities, 
increased  by  10",  will  be  the  moon's  true  distance  from  the  North 
Pole  of  the  Ecliptic.  The  difference  between  this  distance  and 
90°  will  be  the  Moon's  true  Latitude ;  which  will  be  North  01 
South,  according  as  the  distance  is  less  or  greater  than  90°. 

For  the  Equatorial  Parallax. 
With  the  corrected  arguments,  Evection,  Anomaly,  and  Varia- 


342  ASTRONOMICAL  PROBLEMS 

tion,  take  out  the  corresponding  quantities  from  Tables  LXL 
LXII,  and  LXIII.  Their  sum,  increased  by  7",  will  be  the  Equa- 
torial Parallax 

For  the  Semi-diameter. 

With  the  Equatorial  Parallax  as  an  argument,  take  out  the 
moon's  semi-diameter  from  Table  LXV. 

For  the  Hourly  Motion  in  Longitude. 

With  the  arguments  2,  3,  4,  5,  and  6  of  Longitude,  rejecting  the 
two  right-hand  figures  in  each,  take  the  corresponding  equations 
of  the  hourly  motion  in  longitude  from  Table  LXVII.  Find  the 
sum  of  these  equations  and  the  constant  3",  and  with  this  sum  at 
the  top,  and  the  corrected  argument  of  the  Evection  at  the  side, 
take  the  corresponding  equation  from  Table  LXIX  ;  also  with  the 
corrected  argument  of  the  Evection  take  the  corresponding  equa 
tion  from  Table  LXVIII. 

Add  these  equations  to  the  sum  just  found,  and  with  the  result- 
ing sum  at  the  top,  and  the  corrected  anomaly  at  the  side,  take  the 
corresponding  equation  from  Table  LXX  ;  also  with  the  corrected 
anomaly  take  the  corresponding  equation  from  Table  LXXI. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the  re- 
sulting sum  at  the  top,  and  the  corrected  argument  of  the  Variation 
at  the  side,  take  the  corresponding  equation  from  Table  LXXII. 
With  the  corrected  argument  of  the  Variation,  take  the  correspond- 
ing equation  from  Table  LXXIII. 

Add  these  two  equations  to  the  sum  last  found,  and  with  the  re- 
sulting sum  at  the  top,  and  the  argument  of  the  Reduction  at  the 
side,  take  the  corresponding  equation  from  Table  LXXIV.  Also, 
with  the  argument  of  the  Reduction  take  the  corresponding  equa- 
tion from  Table  LXXV.  These  two  equations,  added  to  the  last 
sum,  will  give  the  sum  of  the  principal  equations  of  the  hourly 
motion  in  longitude,  and  the  constants  subtracted  for  the  others 
To  this  add  the  constant  27'  24". 0,  and  the  result  will  be  the 
Moon's  Hourly  Motion  in  Longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  argument  I  of  Latitude,  take  the  corresponding  equa- 
tion from  Table  LXXIX.  With  this  equation  at.  the  side,  and  the 
sum  of  all  the  equations  of  the  hourly  motion  in  longitude,  except 
the  last  two,  at  the  top,  take  the  corresponding  equation  from  Ta- 
ble LXXXI.  With  the  argument  II  of  Latitude,  take  the  corre- 
sponding equation  from  Table  LXXXII.  And  with  this  equation 
at  the  side,  and  the  sum  of  all  the  equations  of  the  hourly  motion 
in  longitude,  except  the  last  two,  at  the  top,  take  the  equation  from 
Table  LXXXIII.  Find  the  sum  of  these  four  equations  and  the 


TO  FIND  THE  MOON*S  LONGITUDE,  ETC.  343 

constant  I".  To  the  resulting  sum  apply  the  constant  —  237".2. 
The  difference  will  be  the  Moon's  true  Hourly  Motion  in  Latitude. 
The  moon  will  be  tending  North  or  South,  according  as  the  sign 
is  positive  or  negative 

Note.  The  errors  of  the  results  obtained  by  the  foregoing  rules, 
occasioned  by  the  neglect  of  the  smaller  equations,  cannot  exceed 
for  the  longitude  15",  for  the  latitude  8V,  for  the  parallax  7",  for 
the  hourly  motion  in  longitude  5",  and  for  the  hourly  motion  in 
latitude  3"  ;  and  they  will  generally  be  very  much  less.  When 
greater  accuracy  is  required,  take  from  Tables  XXXV  to  XXXIX 
the  arguments  from  21  to  31,  along  with  those  from  1  to  20,  and 
their  variations.  The  sums  of  the  numbers  for  these  different  ar- 
guments, respectively,  will  be  the  arguments  of  eleven  small  addi- 
tional equations  of  longitude.  Also,  take  from  the  same  tables  the 
arguments  entitled  XI  and  XII,  along  with  those  in  the  preceding 
columns.  Retain  the  right-hand  figure  of  the  sum  in  column  1  of 
arguments,  and  conceive  a  cipher  to  be  annexed  to  each  number 
in  the  columns  of  arguments  of  Table  XLI.  The  numbers  in  the 
columns  entitled  Diff.for  10,  will  then  be  the  differences  for  a  va- 
riation of  100  in  the  argument. 

For  the  Longitude.  With  the  arguments  21  to  31,  take  the  cor 
responding  equations  from  Tables  XLVII  and  XLVIII,  and  place 
them  in  the  same  column  with  the  equations  taken  out  with  the 
arguments  1,  2,  &c.  to  20.  Take  also  equation  32  from  Table 
XLIX,  as  before.  Find  the  sum  of  the  whole,  (omitting  the  con- 
stant 55",)  and  then  continue  on  as  above.  The  longitude  from 
the  mean  equinox  being  found,  take  the  lunar  nutation  in  longitude 
from  Table  LIV,  and  the  solar  nutation  answering  to  the  given 
date  from  Table  XXVII.  Apply  them  both,  according  to  their 
sign,  to  the  longitude  from  the  mean  equinox,  and  the  result  will 
be  the  more  exact  longitude  from  the  apparent  equinox,  required. 

For  the  Latitude.  With  the  arguments  XI  and  XII,  take  the 
corresponding  equations  from  Table  LIX.  Add  these  with  the 
other  equations,  and  omit  the  constant  10".  The  difference  be- 
tween the  sum  and  90°  will  be  the  more  exact  latitude. 

For  the  Equatorial  Parallax.    With  the  arguments  1,  2,  4,  5, 

6,  8,  9,  12,  13,  take  the  corresponding  equations  from  Table  LX. 
Find  the  sum  of  these  and  the  other  equations,  omitting  the  con- 
stant 7",  and  it  will  be  the  more  exact  value  of  the  Parallax. 

For  the  Hourly  Motion  in  Longitude.     With  the  arguments  1, 

7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  and  18,  of  longitude,  along 
with  the  arguments  2,  3,  4.  5,  and  6,  heretofore  used,  take  the  cor- 
responding equations  firi-ji  Table  LXVII.     Find  the  sum  of  the 


344  ASTRONOMICAL  PROBLEMS. 

whole,  omitting  the  constant  3",  and  proceed  as  in  the  rule  already 
given. 

To  obtain  the  motion  in  longitude  for  the  hour  which  precedes 
or  follows  the  given  time,  with  the  arguments  of  Tables  LXX, 
LXXII,  and  LXXIV,  take  the  equations  from  Tables  LXXV1 
and  LXXVII.  Also,  with  the  arguments  of  Evection,  Anomaly, 
Variation,  and  Reduction,  take  the  equations  from  Table  LXXVIII. 
Find  the  sum  of  all  these  equations.  Then,  for  the  hour  which  fol- 
lows the  given  time,  add  this  sum  to  the  hourly  motion  at  the  given 
time  already  found,  and  subtract  2".0 ;  for  the  hour  which  pre- 
cedes, subtract  it  from  the  same  quantity,  and  add  2".0. 

It  will  expedite  the  calculation  to  take  the  equations  of  the  sec- 
ond order  from  the  tables  at  the  same  time  with  those  of  the  first 
order  which  have  the  same  arguments. 

For  the  Hourly  Motion  in  Latitude.     The  moon's  hourly  mo 
tion  in  latitude  may  be  had  more  exactly  by  taking  with  the  argu- 
ments of  Latitude  V,  VI,  &c.  to  XII,  the  corresponding  equations 
from  Table  LXXX,  and  finding  the  sum  of  these  and  the  other 
equations  of  the  hourly  motion  in  latitude. 

To  obtain  the  moon's  motion  in  latitude  for  the  hour  which  pre 
cedes  or  follows  the  given  time,  with  the  Argument  I  of  Latitude, 
take  the  equation  from  Table  LXXXIV,  and  with  this  equation 
and  the  sum  of  all  the  equations  of  the  hourly  motion  in  longitude 
except  the  last  two,  take  the  equation  from  Table  LXXXV.  Find 
the  sum  of  these  two  equations.  Then,  for  the  hour  which  follows 
the  given  time,  add  this  sum  to  the  Hourly  Motion  in  Latitude  al- 
ready found,  taken  with  its  sign,  and  subtract  1".3 ;  and  for  the 
hour  which  precedes,  subtract  it  from  the  same  quantity,  and  add 
1".3. 

It  will  also  be  more  exact  to  enter  Table  LXXXI  with  the  sum 
of  all  the  equations  of  Tables  LXXIX  and  LXXX,  diminished 
by  1",  instead  of  the  equation  of  Table  LXXIX,  for  the  argument 
at  the  side.  The  numbers  over  the  tops  of  the  columns  in  Table 
LXXXI  are  the  common  differences  of  the  consecutive  numbers 
in  the  columns.  The  numbers  in  the  last  column  are  the  common 
differences  of  the  consecutive  numbers  in  the  same  horizontal  line. 

Exam.  1.  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and  lati- 
tude, on  the  14th  of  October,  1838,  at  6h.  54m.  34s.  P.  M.  mean 
time  at  New  York. 

Mean  time  at  New  York,  October,         14d-   6h-  54m-  348- 
Diif.  of  Long 4    56      4 

Mean  time  at  Greenwich,  October,        14    11    50    38 


TO  FIND  THE  MOON  S  LONGITUDE,  ETC. 


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TO  FIND  THE  MOON*S  REDUCED  PARALLAX,  ETC.  349 

Exam.  2.  Required  the  moon's  longitude,  latitude,  equatorial 
parallax,  semi-diameter,  and  hourly  motions  in  longitude  and  lati- 
tude, on  the  9th  of  April,  1838,  at  8h.  58m.  19s.  P.  M.  mean  time 
at  Washington. 

Ans.  Long.  6s-  19°  45'  31".2;  lat.  36'  21".9  S. ;  equat.  par. 
54'  36".3  ;  semi-diameter  14'  52"  .7 ;  hor.  mot.  in  long.  30'  15"  2 ; 
and  hor.  mot.  in  lat.  2'  47" .0,  tending  south.* 


PROBLEM  XV. 

The  Moon's  Equatorial  Parallax,  and  the  Latitude  of  a  Place, 
being  given,  to  find  the  Reduced  Parallax  and  Latitude. 

With  the  latitude  of  the  place,  take  the  reductions  from  Table 
LXIV,  and  subtract  them  from  the  Parallax  and  Latitude. 

Exam.  1.  Given  the  equatorial  parallax  55'  15",  and  the  lati- 
tude of  New  York  40°  42'  40"  N.,  to  find  the  reduced  parallax  and 
latitude. 

Equatorial  parallax,     ...         55'  15" 
Reduction, 5 

Reduced  parallax,       .         .        .         55  10 

Latitude  of  New  York,        .  40°  42'  40/;  N. 

Reduction,          .         .         .         .          11  20 


Reduced  Lat.  of  New  York,          40    31    20 

2.  Given  the  equatorial  parallax  60'  36",  and  the  latitude  of 
Baltimore  39°  17'  23"  N.,  to  find  the  reduced  parallax  and  latitude. 

Ans.    Reduced  par.  60'  32",  and  reduced  lat.  39°  6'  9". 

3.  Given  the  equatorial  parallax  57'  22",  and  the  latitude  of 
New  Orleans  29°  57'  45"  N.,  to  find  the  reduced  parallax  and  lat- 
itude. 

Ans.  Reduced  par.  57'  19",  and  reduced  lat.  29°  47'  50". 


PROBLEM  XVI. 

To  find  the  Longitude  and  Altitude  of  the  Nonagesimal  Degree 
of  the  Ecliptic,  for  a  given  time  and  place. 

For  the  given  time,  reduced  to  mean  time  at  Greenwich,  find  the 
sun's  mean  longitude  and  the  argument  N  from  Tables  XVIII, 
XIX,  XX,  and  XXI.  To  the  sun's  mean  longitude,  apply  accord- 
ing to  '.is  sign  the  nutation  in  right  ascension,  taken  from  Table 

*  The  smaller  equations  were  omitted  in  working  this  example. 


350  ASTRONOMICAL  PROBLEMS. 

XXVII  with  argument  N ;  and  the  result  will  be  the  right  ascen- 
sion of  the  mean  sun,  (see  Art.  127,)  reckoned  from  the  true  equi- 
nox. 

Reduce  the  mean  time  of  day  at  the  given  place,  expressed  as- 
tronomically, to  degrees,  &c.,  and  add  it  to  the  right  ascension  oi 
the  mean  sun  from  the  true  equinox.  The  sum,  rejecting  360° 
when  it  exceeds  that  quantity,  will  be  the  right  ascension  of  the 
midheaven,  or  the  sidereal  time  in  degrees. 

Next,  find  the  reduced  latitude  of  the  place  by  Problem  XV ; 
and  when  it  is  north,  subtract  it  from  90° ;  but  when  it  is  south, 
add  it  to  90°.  The  sum  or  difference  will  be  the  reduced  distance 
of  the  place  from  the  north  pole. 

Also,  take  the  obliquity  of  the  ecliptic  for  the  given  year  from 
Table  XXII.* 

These  three  quantities  having  been  found,  the  longitude  and  alti- 
tude of  the  nonagesimal  degree  may  be  computed  from  the  follow- 
ing formulae : 

log.  cos  i  (H  -  w)  -  log.  cos  i  (H  +  w)  =  A  .  .  .  (1); 
log.  tang  i  (H  -  w)  +  10  -  log.  tang  1  (H  +  w)  =  B  .  .  .  (2) ; 
log.  tang  E  =  A  +  log.  tang  £  (S  —  90°)  .  .  .  (3) ; 
log.  tang  F  =  log.  tang  E  +  B  .  .  .  (4) ; 
N  =  E4-F  +  90°  .  .  .  (5); 

log.  tang  \h  —  log.  cos  E  +  log.  tang  j  (H  +  w)  +  ar.  co.  log 

cos  F  -  20  ...  (6). 
in  which 

H  =  the  reduced  distance  of  the  place  from  the  north  pole ; 

w  =  the  Obliquity  of  the  Ecliptic ; 

S  =  the  Sidereal  Time  converted  into  degrees  ; 

N  =  the  required  Longitude  of  the  Nonagesimal ; 

h  =  the  required  Altitude  of  the  Nonagesimal ; 

E  and  F  are  auxiliary  angles. 

We  first  find  the  logarithmic  sums  A  and  B.  With  these  we  de- 
termine the  angles  E  and  F  by  formulae  (3)  and  (4),  and  with  these 
again  N  and  h  by  formulae  (5)  and  (6). 

The  angles  E,  F,  are  to  be  taken  less  than  180°;  and  less  or 
greater  than  90°,  according  as  the  sign  of  their  tangent  proves  to  be 
positive  or  negative. 

Note  1 .  In  case  the  given  place  lies  within  the  arctic  circle,  we 
must  take,  in  place  of  formula  (5),  the  following : 
N  =  E  -  F  +  90°. 

*  If  great  precision  is  required,  the  apparent  obliquity  is  to  be  used  b  place  of  th* 
mean.  (See  Prob.  X.) 


TO  FIND  THE  LONG.  AND  ALT.  OF  NONAGESIMAL  DEGREE.  351 

Note  2.  As  the  obliquity  of  the  ecliptic  varies  but  slowly  from 
year  to  year,  the  values  which  have  once  been  found  for  the  loga- 
rithms A,  B,  and  log.  tang  \  (H  +  w)  (C),  will  answer  for  several 
years  from  the  date  of  their  determination,  unless  very  great  accu- 
racy is  required. 

Note  3.  The  angle  h  derived  from  formula  (6),  is  the  dis- 
tance of  the  zenith  of  the  given  place  from  tbe  north  pole  of  the 
ecliptic.  This  is  not  always  equal  to  the  altitude  of  the  nonagesi- 
mal  Throughout  the  southern  hemisphere,  and  frequently  in  the 
northern  near  the  equator,  it  is  the  supplement  of  the  altitude.  In 
employing  this  angle  in  the  following  Problem,  it  is,  however,  for  the 
sake  of  simplicity,  called  the  altitude  of  the  nonagesimal  in  all  cases. 

Exam.  1 .  Required  the  longitude  and  altitude  of  the  nonagesi- 
mal degree  of  the  ecliptic  at  New  York,  on  the  18th  of  September, 
1838,  at  3h.  52m.  56s.  P.  M.  mean  time. 

The  sun's  mean  longitude  taken  from  the  tables,  for  the  given 
time,  is  5s-  27°  19'  17",  and  the  argument  N  is  987.  The  nutation 
taken  from  Table  XXVII  with  argument  N  is  —  1".  Hence,  the 
right  ascension  of  the  mean  sun,  reckoned  from  the  true  equinox, 
is  5s-  27°  19'  16".  The  given  time  of  day,  expressed  astronomi- 
cally, is  3h.  52m.  56sec. ;  which  in  degrees  is  58°  14'  0". 

The  reduced  latitude  of  New  York,  found  by  Problem  XV,  is 
40°  31'  20",  and  this  taken  from  90°  leaves  the  polar  distance  49° 
28'  40".  The  obliquity  of  the  ecliptic,  derived  from  Table  XXII, 
is  23°  27'  37". 

Given  time  in  degrees,     .        .        .     58°  14'    0" 
R.  Asc.  of  mean  sun,        .        .         .  177    19  16 

Sidereal  time  in  degrees  (S),     .         .  235    33  16 

90 


2)145    33  16 
H        .     .   49°28/40" 
w          ..   23  27  37  i  (S  —  90)  72    46  38 

Diff     .     .   26     1     3 
Sum    .     .    72  56  17 

|diff.  .     .13     0  31     .      cos.  9.98870     .     tan.  +  10,19.36366 
sum  .     .   36  28     8     .      cos.  9.90535     .     tan.       C.  9.86871 


A.  0.08335  B.  9.49495 

i(S  -  90°)  72  46  38  .   tan.  0.50866 

E    .    75  38  55  .   tan.  0.59201  .  cos.     9.39422 

B.  9.49495  C.  9.86871 

F    .    50  41  55  .   tan  0.08696  .  ar.  co.  cos.  0.19832 
90  0  0 

i  alt.  non.  16°  7'  54"  .     tan.  9.46125 


long.  non.    216  20  50 


alt  non.  32  15  48 


352  ASTRONOMICAL  PROBLEMS. 

2.  Required  the  longitude  and  altitude  of  the  nonagesaimal  de- 
gree of  the  ecliptic  at  New  York,  on  the  10th  of  May,  1838,  at 
llh.  33m.  56sec.  P.  M.  mean  time. 

Ans.  Long.  200°  12'  23",  and  alt.  37°  (X  34". 

PROBLEM  XVII. 

To  find  the  Apparent  Longitude  and  Latitude,  as  affected  by 
Parallax,  and  the  Augmented  Semi-diameter  of  the  Moon  ;  the 
Moon's  True  Longitude,  Latitude,  Horizontal  Semi-diameter, 
and  Equatorial  Parallax,  and  the  Longitude  and  Altitude  of 
the  Nonagesimal  Degree  of  the  Ecliptic,  being  given. 

We  have  for  the  resolution  of  this  Problem  the  following  for 
mulae : 

log.  x  =  log.  P  + log.  cos  ft+ar.  co.log.cos  X—  10  ...  (1); 

c  =  log.  x  +  log.  tang  h  —  10  ...  (2) ; 
log.  u  —  c  +  log.  sin  K  —  10  ...  (3) ; 
log.tt'  =  c+log.  sin(K  +  w)-  10  .  .  .  (4); 
log.p=c+log.  sin(K  +  w')-  10  .  .  .  (5); 
Appar.  long.  =  true  long.  +p  .  .  .  (6) ; 

log.  tang  X'  =  log.  p  +  ar.  co.  log.  cos  X  +  ar.  co.  log.  u  +  log. 

sin  (X-  x)  -10*  ...  (7); 

log.  v  =  log.  P  +  log.  cos  h  +  log.  cos  X'  —  10   .  .  .  (8) ; 
log.  z  =  log.  v  +  log.  tang  h  +  log.  tang  X'  -f-  log.  cos 

(K  +  *p)-30.  .  .  (9); 
*=v_£  .  .  .  (10); 
Appar.  lat.     =  true  lat.  —  «  .  .  .  (11); 

log.  R'  =  log.  p  +  ar.  co.  log.  cos  X  +  ar.  co.  log.  u  +  log 

cos  X'  -flog.  R  —  10  ...  (12)  : 
in  which 

P  =  the  Reduced  Parallax  of  the  Moon ; 
h  =  the  Altitude  of  the  Nonagesimal ; 
X  =  the  True  Latitude  of  the  Moon  (minus  when  south) ; 
K  =  the  Longitude  of  the  Moon,  minus  the  longitude  of  the  No- 
nagesimal ; 

p  =  the  required  Parallax  in  Longitude  ; 
X;  =  the  approximate  Apparent  Latitude  of  the  Moon ; 
if  =  the  required  Parallax  in  Latitude ; 
R  =  the  True  Semi-diameter  of  the  Moon ; 
R'  =  the  Augmented  Semi-diameter  of  the  Moon ; 
xt  u,  u',  v,  z,  are  auxiliary  arcs. 

*  Formula  (7)  will  be  rendered  more  accurate  by  adding  to  it  the  ar.  co.  cos 
X  — 10,  and  will  generally  give  the  apparent  latitude  with  sufficient  accuracy; 
thus  rendering  formulae  (8),  (9),  (10),  and  (11)  unnecessary. 


AND  LAT.  353 

Formulae  (1),  (2),  (3),  (4),  and  (5),  being  resolved  in  succession, 
we  derive  the  apparent  longitude  from  formula  (6) ;  then  the  appa- 
rent latitude  from  equations  (7),  (8),  (9),  (10),  (11);  and  lastly, 
the  augmented  semi-diameter  from  equation  (12.) 

The  latitude  of  the  moon  must  be  affected  with  the  negative 
sign  when  south ;  and  the  apparent  latitude  will  be  south  when  it 
comes  out  negative.  In  performing  the  operations,  it  is  to  be  re- 
membered that  the  cosine  of  a  negative  arc  has  the  same  sign  as 
the  cosine  of  a  positive  arc  of  an  equal  number  of  degrees  ;  but 
that  the  sine  or  tangent  of  a  negative  arc  has  the  opposite  sign  from 
the  sine  or  tangent  of  an  equal  positive  arc.  Attention  must  also 
be  paid  to  the  signs  in  the  addition  and  subtraction  of  arcs.  Thus, 
two  arcs  affected  with  essential  signs,  which  are  to  be  added  to 
each  other,  are  to  be  added  arithmetically  when  they  have  like 
signs,  but  subtracted  if  they  have  unlike  signs  ;  and  when  one  arc 
is  to  be  taken  from  another,  its  sign  is  to  be  changed,  and  the  two 
united  according  to  their  signs.  An  arithmetical  sum,  when  taken, 
will  have  the  same  sign  as  each  of  the  arcs ;  and  an  arithmetical 
difference  the  same  sign  as  the  greater  arc. 

The  use  of  negative  arcs  may  be  avoided,  though  the  calculation 
would  be  somewhat  longer,  by  using  the  true  polar  distance  d,  and 
the  approximate  apparent  polar  distance  d',  in  place  of  X  and  X7, 
substituting  sin  d  for  cos  X,  cos  (d  -h  x)  for  sin  (X  —  a?),  -sin  d'  for 
cos  X7,  log.  co-tang  d1  for  log.  tang  X' ;  and  observing  that  p  is 
to  be  subtracted  from  the  true  longitude  in  case  the  longitude  of 
the  nonagesimal  exceeds  the  longitude  of  the  moon  ;  that  z,  when 
it  comes  out  negative,  is  to  be  added  to  u,  which  is  always  positive 
to  the  north  of  the  tropic,  otherwise  subtracted  ;  and  that  the  par- 
allax in  latitude  is  to  be  applied  according  to  its  sign  to  the  true 
polar  distance. 

In  seeking  for  the  logarithms  of  the  trigonometrical  lines,  it  will 
be  sufficient  to  take  those  answering  to  the  nearest  tens  of  seconds. 

Note  1.  When  great  accuracy  is  not  desired,  u'  may  be  taken 
for  p,  from  which  it  can  never  differ  more  than  a  fraction  of  a 
second. 

Note  2.  In  solar  eclipses  the  moon's  latitude  is  very  small,  and 
formula  (7)  may  be  changed  into  the  following : 

log.  X'  =  log.  p  +  ar.  co.  log.  cos  X -f  ar.  co.  log.  u  +log.  (X  —  #)— 10 

and  cos  X7  omitted  in  formula  (12)  without  material  error. 

Formulae  (8),  (9),  (10),  and  (11),  may  also  now  be  dispensed 
with,  unless  very  great  precision  is  desired,  and  the  value  of  X' 
given  by  the  above  formula  taken  for  the  apparent  latitude. 

It  is  to  be  observed  also,  that  in  eclipses  of  the  sun  P  is  taken 
equal  to  the  reduced  parallax  of  the  moon  minus  the  sun's  horizon- 
tal parallax.  By  this  the  parallax  of  the  sun  in  longitude  and  lati- 
tude is  referred  to  the  moon,  and  the  relative  apparent  places  of 
the  sun  and  moon  are  correctly  obtained,  without  the  necessity  of 

23 


854  ASTRONOMICAL  PROBLEMS. 

a  separate   computation  of  the  sun's  parallax  in  longitude   and 
latitude. 

Exam.  1.  About  the  time  of  the  middle  of  the  occultation  of  the 
star  Antares,  on  the  10th  of  May,  1838,  the  moon's  longitude,  by 
the  Connaissance  des  Terns,  was  247°  37'  6".7;  latitude  4°  14 
14'. 7  S. ;  semi-diameter  15'  24".2;  and  equatorial  parallax  56 
31  ".7;  and  the  longitude  of  the  nonagesimal  at  New  York  was 
200°  12'  23"  ;  the  altitude  37°  0'  34"  ;  required  the  apparent  Ion 
gitude  and  latitude,  and  the  augmented  semi-diameter  of  the  moon 
at  New  York,  at  the  time  in  question. 

Equat.  par.     56' 31  ".7  Moon's  long.     247°  37'    7" 

Reduction  4  .6  Long,  nonag.    200  12  23 

P  =  56  27  .1  K=  47  24  44 

h  =  37  0  34 
X  =  -4  14  14.7 

P  3387".!     .    .  log.  3.52983 

h  .37°  0'34"     .    .  cos.  9.90230 

a.  3.43213 
X    .    .     —41415       ar.  co.  cos.  0.00119 

x    .    .    .     45  12  .  2712"   .  log.  3.43332 
h    .    .    .  37  0  34  .    .      tan.  9.87725 


c.  3.31057 
47  24  44  .         sin.  9.86701 


25  5  .  1505"   .  log.  3.17758 

c.  3.31057 
ti    .    .  47  49  49  .    .    .  sin.  9.86991 

25  15  .  1515".2  .  log.  3.18048 

c.     3.31057 
47   49  59  .  sin.  9.86993 


p  25  15.3  .  1515".3  .     log.  3.18050 

True  long.  .        .  247    37    6.7 

Appar.  long.         .  248      2  22.0 

p log.  3.18050 

x-a?   .         .         .    -4    59  27     .         .         .     sin.  8.93957- 

x ar.  co.  cos.  0.00119 

tt         •  ar.  co.  log.  6.82242 

*'          ...    -6     1   10  tan.  8.94368- 


TO  FIND  THE  MOON'S  APPAR.  LONG.  AND  LAT.  355 

X'  .5°  1'  10"  .    .    .  cos.  9.99833 

a.  3.43213 


t>  44  54.4  .  2694".4  .  log.  3.43046 

h    . tan.  9.87725 

X'  ?;-; tan.  8.94368— 

.  47  37  22  ...  cos.  9.82867 


z    .    .    .    — 2  0.2  .  120".2  .  log.  2.08006 - 

v-z  .    .    .     46  54.6 

v—z  (sign  changed)   —46  54.6 

True  lat.   .    .  -4  14  14.7 


Appar.  lat  . 

P 

X 

5  1 

9.3 

S. 

•  log. 

CO.  COS. 

co.  log. 

.   COS. 

.  log. 

3.18050 
0.00119 
6.82242 
9.99833 
2.96577 

u 
X' 

R   .   . 

•   • 
*  15 

• 
24.2 

* 

.  ar. 
924".2 

Augm.  semi-diam.  15  29.4  .    929".4  .     log.  2.96821 

Exam.  2.  About  the  middle  of  the  eclipse  of  the  sun  on  the  18th 
of  September,  1838,  the  moon's  longitude  was  175°  29'  19".0, 
latitude  47'  47".5,  equatorial  parallax  53'  53".5,  and  semi-diame- 
ter 14'  4 I'M  ;  and  the  longitude  of  the  nonagesimal  at  New  York 
was  216°  20'  50",  the  altitude  32°  15'  48":  required  the  apparent 
longitude  and  latitude,  and  the  augmented  semi-diameter  of  the 
moon. 

Equal,  paral.     53'  53".5  Moon's  long.     1 75°  29'  1 9*' 

Reduction,  4  .4  Long,  nonag.    216  20  50 


53  49  .1 

K  =  -40  51  31 

Sun's 

paral.      8  .6 

h  =  32  15  48 

P=53  40  .5 

X  =   0  47  47.5 

P 

.  3220".5  . 

log.  3.50792 

h 

32°  15'  48"  . 

cos.  9.92716 

X 

47  47.5  . 

ar.  co.  cos.  0.00004 

X 

45  23.5  .  2723". 

5  .    log.  3.43512 

k 

32  15  48   . 

tan.  9.80023 

c.  3.23535 

K 

.  -40  51  31   . 

sin.  9.81570  — 

—1845       .1125"      .         log.  3.05L05- 


356 


ASTRONOMICAL   PROBLEMS. 


K-t-w  . 
u' 


—41°  ICV  16" 
-18  52.9 


K-H*'. 

—41   10  24       . 

P 

True  long.    . 

Appar.  long. 

P 
X 

—  18  52.9    .  1132".9 
175  29  19.0 

175  10  26.1 

£ 

u 
X  —  x    . 

2'  24".0    .  "  144;/.0 

Appar.  latitude      2' 24".9  N.    144".9 


X 
u 
R 


c.   3.23535 
sin.  9.81844— 

1132".9  .    log.  3.05379- 


c.  3.23535 
sin.  9.81844— 

log.  3.05379— 


,  log.  3.05379 
ar.  co.  cos.  0.00004 
ar.  co.  log.  6.94895 
log.  2.15836 

•&l   .  log.  2.16114 

,  *   .  log.  3.05379 

ar.  co.  cos.  0.00004 

ar.  co.  log.  6.94895 

.  log.  2.94502 


Augm.  semi-diam.  14  46  .7  .  886".7  .    .  log.  2.94780 


PROBLEM   XVIII. 

To  find  the  Mean  Right  Ascension  and  Declination,  or  Longitude 
and  Latitude  of  a  Star,  for  a  given  time,  from  the  Tables. 

Take  the  difference  between  the  given  year  and  1840.  Then 
seek  in  Table  XV  for  the  fraction  of  the  year  answering  to  the 
given  month  and  days,  and  add  it  to  this  difference,  if  the  given 
time  is  after  the  beginning  of  the  year  1 840 ;  but  if  it  is  before, 
subtract  it.  Multiply  the  sum  or  difference  by  the  annual  variation 
given  in  the  catalogue,  (Table  XC,  or  XCII,)  and  the  product  will 
be  the  variation  in  the  interval  between  the  given  time  and  the 
epoch  of  the  catalogue.  Apply  this  product  to  the  quantity  given 
in  the  catalogue,  according  to  its  sign,  if  the  given  time  is  after  the 
beginning  of  the  year  1840,  but  with  the  opposite  sign  if  it  is  before, 
and  the  result  will  be  the  quantity  sought.  (See  Prob.  XXL  Note.} 

Exam.  1.  Required  the  mean  right  ascension  and  decimation  of 
the  star  Sirius  on  the  15th  of  August,  1842. 

Interval  between  given  time  and  beginn.  of  1840,  (tt)       2.619yrs. 
Annual  variation  of  right  ascension,          .         .         .  2.646s. 

Variation  of  right  ascension  for  interval  t,         .         .  6.93s. 


TO  FIND  A  STAR'S  ABERR.  IN  RIGHT  ASCENSION,  ETC.      357 

A  similar  operation  gives  for  the  variation  of  decimation  in  the 
same  interval,  11  ".65. 

Mean  right  ascen.,  beginning  of  1840,  Table  XC,     6^  38"-  5.76* 
Variation  for  interval  tt     .         .         .         ...  +  6.93 


Moan  right  ascension  required,          .         ,  6  38  12.69 

Mean  decimation, beginning  of  1840,          .         .  16°  30'    4".79S. 
Variation  for  interval  t,     .         .         .        ..         .  + 1 1  .65 


Mean  declination  required,        .         .         .         .  16  30  16  .44  S. 

2.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  20th  of  October,  1838. 

Interval  between  given  time  and  begin,  of  1840,  (t)          1.200yrs. 
Annual  variation  of  longitude,         ....         50".210 

Variation  of  longitude  for  interval  t,  60". 2 

A  similar  operation  gives  for  the  variation  of  latitude  in  the  same 
interval  0".4. 

Mean  longitude,  beginning  of  1840,          .         2'-   7°    33'    5".9 
Variation  for  interval  t,  ...  —  1     0  .2 


Mean  longitude  required,        .         .         .         2     7     32     5  .7 

Mean  latitude, beginning  of  1840,    .         .  5°    28'  38".0  S. 

Variation  for  interval  t,  .         .         ,  +  0  .4 


Mean  latitude  required,  .         .         .         .  5     28  38  .4  S. 

3.  Required  the  mean  right  ascension  and  declination  of  Capella 
on  the  9th  of  February,  1839  ? 

Ans.  Mean  right  ascension  5h-  4m-  48.74"-,  and  mean  declination 
45°  49'  38".53  N. 

4.  Required  the  mean  longitude  and  latitude  of  Aldebaran  on 
the  1 6th  of  April,  1845? 

Ans.  Mean  longitude  2s-  7°  37'  31".4,  and  mean  latitude  5°  2& 
36".2. 


PROBLEM  XIX. 

To  find  the  Aberrations  of  a  Star  in  Eight  Ascension  and  Declina- 
tion, for  a  given  Day.     (See  Prob.  XXI.  Note.) 

This  problem  may  be  resolved  for  any  of  the  stars  in  the  cata- 
logue of  Tab'.e  XC  by  means  of  the  following  formulae  : 


358  ASTRONOMICAL  PROBLEMS. 

log.  (aber.  in  right  ascen.)  =  M  +  log.  sin  (0+9)  —  10. 
log.  (aber.  in  declin.)          =  N  +  log.  sin  (O  +  d)  —  10, 

in  which  M,  N,  are  constant  logarithms,  O  the  longitude  of  the  sun 
on  the  given  day,  and  9,  6,  auxiliary  angles.  M,  N,  and  the  an- 
gles 9,  6,  are  given  for  each  of  the  stars  in  the  catalogue,  in 
Table  XCI.  O  may  be  derived  from  an  ephemeris  of  the  sun, 
or  it  may  be  computed  from  the  solar  tables  by  Problem  IX. 

Exam.  1 .  What  was  the  amount  of  aberration,  in  right  ascen- 
sion and  declination,  of  a  Orionis  on  the  20th  of  December,  1837, 
the  sun's  longitude  on  that  day  being  8s-  28°  28'  1 

Right  Ascension. 
Table  XCI,  9      -        6s-    3°  13'     M  .       .        0.1361 

O     .         8    28    28 


O  +  9.3      1    41          .         .sin.  9.9998 

Aberration  =  1".37     .         .         .         .log.  0.1359 

Declination. 

Table  XCI,  0        .         8s- 28°  23'     N.        .        0.7521 
O      .        8   28   88 


O-H  •     5   26   51          .         .sin.  8.7399 

Aberration  =  0".31     ....  log.  T.4920 

2.  Required  the  aberrations  in  right  ascension  and  declination 
of  a  Andromedae  on  the  1st  of  May,  1838,  the  sun's  longitude  be- 
ing I8  10°  38'. 

Ans.  Aberr.  in  right  ascension  —  1".07,  and  aberr.  in  declina- 
tion -  11".69. 


PROBLEM  XX. 

Tojind  the  Nutations  of  a  Star  in  Right  Ascension  and  Declina- 
tion, for  a  given  Day. 

This  Problem  may  be  solved  by  means  of  the  formulae, 

log.  (nuta.  in  right  asc.)  =  M7  H-log.  sin  (Q>  +9')  —  10 ; 
log.  (nuta.  in  declin.)       =  N'  +  log.  sin  (&  +  6')  —  10 ; 

111  which  M',  N',  are  constant  logarithms,  &  the  mean  longitude  of 
the  moon's  ascending  node,  and  9',  6',  auxiliary  angles.  M',  N', 
and  the  angles  9',  0',  are  given  for  each  of  the  stars  in  the  cata- 
logue, in  Table  XCI.  The  mean  longitude  of  the  moon's  ascend- 
.ng  node  is  given  for  every  tenth  day  of  the  year  in  the  Nautical 
Almanac,  page  242,  and  may  be  easily  found  for  any  intermediate 


TO  FIND  A  STAR'S  NUTATION  IN  RIGHT  ASCEN.,  ETC.      359 

day  from  the  daily  motion  inserted  at  the  foot  of  the  column  of 
longitudes.  It  may  also  be  had  by  finding  the  supplement  of  the 
moon's  node,  for  the  given  time,  from  the  lunar  tables,  and  sub- 
tracting it  from  12s-  0°  7'.  (See  Note  to  Prob.  XXL) 

Exam.  1 .  What  was  the  amount  of  the  nutation,  in  right  ascen 
sion  and  declination,  of  a  Orionis  on  the  20th  of  December,  1837, 
the  mean  longitude  of  the  moon's  node  on  that  day  being  18°  54'  7 

Right  Ascension. 

Table  XCI,  <p'        .     6*-  0°    15'     M'     .        .    0.0481 
ft         .     0  18     54 


ft +9'.     619       9     .         .       sin.  9.5159- 
Nutation  =  -  0".37        .      log.T5640- 

Declination. 

Table  XCI,  f         .     3s-  2°   37'     N'    .        .    0.9657 
&         .     0  18     54 


.     3  21     31      .         .       sin.  9.9686 

Nutation  =       8".60        .      log.  0.9343 

2.  Required  the  nutations  in  right  ascension  and  declination  of 
a  Andromedae  on  the  1st  of  May,  1838. 

Ans.  Nutation  in  right  ascension  —  0".54,  and  nutation  in  de- 
clination -  1".43. 

Note.  When  the  apparent  place  of  a  star  is  desired  with  great 
accuracy,  the  solar  nutations  must  also  be  estimated  and  allowed 
for.  These  may  be  determined  by  repeating  the  process  for  find- 
ing the  lunar  nutations,  only  using  twice  the  sun's  longitude  in 
place  of  the  longitude  of  the  moon's  node,  and  multiplying  the  re- 
sults by  the  decimal  .075. 

The  calculation  of  the  solar  nutations  in  Example  1st,  is  as  fol- 
lows : 

Right  Ascension. 

Table  XCI,  9'    .        .    6s-  0°    15'    M'    .        .    0.0481 
2O     .         .     5  26     56 


2O  +  9'.  11  27     11      .        .       sin.  8.6914— 

—  0".05    .     log.~2~.7395- 
.075 

Solar  Nutat.  =  -  0".00 


360  ASTRONOMICAL  PROBLEMS. 

Declination. 

Table  XCI,  *'    .         .     3s-   2°    37'     N;     .  0.9657 

2O          .     5  26     56 


20+*'.     829     33      .         .     sin.  10.0000— 

—  9".24    .     .        0.9657— 
.075 

Solar  Nutat.  =  -  0".69 

Tn  Example  2d,  we  find  for  the  solar  nutation  in  right  ascension, 
—  0".08,  and  for  the  solar  nutation  in  declination,  —  0".51. 


PROBLEM  XXI. 

To  find  the  Apparent  Right  Ascension  and  Declination  of  a  Star, 
on  a  given  Day. 

Find  the  mean  right  ascension  and  declination  for  the  given  day 
by  Problem  XVIII ;  then  compute  the  aberrations  in  right  ascen- 
sion and  declination  by  Problem  XIX,  and  the  lunar  and  solar  nu- 
tations in  right  ascension  and  declination  by  Problem  XX.  Apply 
the  aberrations  and  nutations  according  to  their  signs,  to  the  mean 
right  ascension  and  declination  on  the  given  day,  observing  that  the 
decimation  when  south  is  to  be  marked  negative,  and  the  results 
will  be  the  apparent  right  ascension  and  declination  sought. 

Exam.  1.  What  was  the  apparent  right  ascension  and  declina- 
tion of  a  Orionis  on  the  20th  of  December,  1837  ? 

h.  m.      s.  °     '      " 

Table  XC,  M.  right  ascen.    5  46  30.71     M.  dec.  7  22  17.14N 
Variations  —  6.59  —2.42 


5  46  24.12  7  22  14.72 

Aberr.   .         .  +1.37  .         .           -fO.31 

Lun.  nutat.    .  —  0.37  .     *  *           +  8.60 

Sol.  nutat.  0.00  -  0.69 


App.  right  asc.  5  46  25.12    App.dec.  7  22  22.94N. 

2.  Required  the  apparent  right  ascension  and  declination  of 
a  Andromedae  on  the  1st  of  May,  1838. 

Ans.  Appar.  right  ascen.  Oh.  Om.  0.90s.,  and  appar.  dec. 
28°  11'39".92. 

NOTE.— V  Prob.  XYIII.  use  Table  XC.  (a)  for  calculations  after  1860.  Table 
XCI.  will  not  give  accurate  results  for  dates  after  1860.  The  method  now  adopted 
in  solving  Prob.  XXI.  is  by  means  of  tables  published  annually  in  theK  Almanac. 


TO  FIND  A  STAR'S  ABERRATION  IN  LONGITUDE,  ETC.       361 


PROBLEM  XXII. 

To  find  the  Aberrations  of  a  Star  in  Longitude  and  Latitude,  for 
a  given  Day. 

The  formulae  for  the  computation  are, 

log.  (aber.  in  long.)  =  1.30880  +  log.  cos  (6s.  +  O  —  L)  +  ar. 

co.  log.  cos  X  —  10 ; 

log.  (aber.  in  lat.)  =  1.30880  +  log.  sin  (6s.  +  O  —  L)  +  log. 

sin  X  —  20  ; 

in  which  O  =  longitude  of  the  sun  on  the  given  day ;  L  =  mean 
longitude  of  the  star ;  and  X  =  mean  latitude  of  the  star. 

Exam.  1 .  Required  the  aberrations  in  longitude  and  latitude  of 
Antares  on  the  26th  of  February,  1838,  the  sun's  longitude  on  thai 
day  being  11s- 7°  29'. 

By  Prob.  XVIII,  L  =  8s-  7°  30',     and  X  =          4°  32'  S 
6s. +  O     .     17     729       Const,  log.    1.3088 

6s.  +  O  -  L   8  29   59    .         .      cos.   6.4637  — 
X      .         .  4   32   .    ar.  co.  cos.   0.0014 

Aberr.  in  long.  =  —  0".00  .  log.  "3.7739  — 

Const,  log.    1.3088 

6s.  +  O  -  L  8'-  29°  59;  .         .      sin.  10.0000  - 
X      ...          4  32    .         .      sin.    8.8978 

Aberr.  in  lat.  =  -  1".61  .  log.    0.2066  - 

2.  Required  the  aberrations  in  longitude  and  latitude  of  Arc- 
turus  on  the  5th  of  October,  1838,  the  sun's  longitude  being 
6s-  11°  47'. 

Ans.  Aberr.  in  long.  —  23".34,  and  aberr.  in  lat.  1".85. 
Note.  The  nutation  in  longitude  of  a  fixed  star  may  be  found 
after  the  same  manner  as  the  nutation  in  longitude  of  the  sun. 
See  Problem  IX.) 


PROBLEM  XXIII. 

To  find  the  Apparent  Longitude  and  Latitude  of  a  Star,  for  a 

given  Day. 

Find  the  mean  longitude  and  latitude  on  the  given  day  by  Prob 
lem  XVIII.   Find  also  the  aberrations  in  longitude  and  latitude  by 
Problem  XXII,  and  the  nutation  in  longitude,  as  in  Problem  IA. 
Apply  the  aberration  and  nutation  in  longitude,  according  to  their 


362  ASTRONOMICAL  PROBLEMS. 

signs,  to  the  mean  longitude,  and  the  result  will  be  the  apparent 
longitude ;  and  apply  the  aberration  in  latitude  according  to  its 
sign,  to  the  mean  latitude,  and  the  result  will  be  the  apparent 
latitude. 

Exam.  1.  Required  the  apparent  longitude  and  latitude  of  An 
tares  on  the  26th  of  February,  1838. 

Table  XC,  M.  long.   8*  7°  31'  45".2         M.  lat.  4°  32'  51". 6  S. 
Var.  —  1  32  .57  0  .78 


8     7  30  12  .63         .      .4  32  50  .82 
Aberr.     .  0  .00        v     v          -  1  .61 

Nutat.  -  4  .40 


App.  long.  8     7  30     8  .23    App.  lat.  4  32  49  .21  S 

2.  Required  the  apparent  longitude  and  latitude  of  Arcturus  0*1 
the  5th  of  October,  1838. 

Ans.  Appar.long.  6s-  21°  58'  37".4,  andappar.  lat.30°  51'  19  M. 

PROBLEM   XXIV. 

To  compute  the  Longitude  and  Latitude  of  a  Heavenly  Body  from 
its  Right  Ascension  and  Declination,  the  Obliquity  of  the  Eclip- 
tic being  given. 

This  Problem  may  be  solved  by  means  of  the  following  for- 
mulae : 

log.  tang  x  =  log.  tang  D  +  ar.  co.  log.  sin  R ; 

log.  tang  L=log.  cos  (x— w) -f- log.  tang  R  +  ar.  co.log.eosa?— 10; 

log.  tang  X  =  log.  tang  (x  —  GO)  +  log.  sin  L  —  10  ; 

in  which 

R  =  the  Right  Ascension  ; 

D  =  the  Declination  (minus  when  South) ; 

L  =  the  Longitude  ; 

X  =  the  Latitude  ; 

w  =  the  Obliquity  of  the  ecliptic ; 

x  is  an  auxiliary  arc.  It  must  be  taken  according  to  the  sign  of 
its  tangent,  but  always  less  than  180°.  The  longitude  is  generally 
in  the  same  quadrant  as  the  right  ascension.  The  latitude  must 
be  taken  less  than  90°,  and  will  be  north  or  south,  according  as  the 
sign  is  positive  or  negative. 

Note.  When  the  mean  longitude  and  latitude  are  to  be  derived 
from  the  mean  right  ascension  and  declination,  the  mean  obliquity 
of  the  ecliptic  is  taken.  When  the  apparent  longitude  and  latitude 
are  to  be  derived  from  the  apparent  right  ascension  and  declina- 
tion, found  as  in  Problem  XXI,  the  apparent  obliquity  is  taken. 


TO  COMPUTE  THE  RIGHT  ASCEN.  AND  DEC.  OF  A  BODY.       363 

The  mean  obliquity  of  the  ecliptic  at  any  assumed  time  is  easily 
deduced  from  Table  XXII.  The  apparent  obliquity  is  found  by 
Problem  X. 

Exam.  1.  On  the  20th  of  June,  1838,  the  right  ascension  of 
Capella  was  76°  11'  29",  the  declination  45°  49'  35"  N.,  and  the 
obliquity  of  the  ecliptic  23°  27'  37"  ;  required  the  longitude  and 
latitude. 

D  =  45°  49'  35"     ...         tan.  0.0125295 
R  =  76   11  29  .      ar.  co.  sin.  0.0127367 


x  =  46  39  56       .         .         .         tan.  0.0252662 

w  =  23  27  37  


x  —  u  =  23  12  19  .  .  .  cos.  9.9633623 
R  =  76  11  29  .  .  .  tan.  0.6094483 
x  =  46  39  56  ar.  co.  cos.  0.1635240 


Long.  =  79   36     4       .         .         .         tan.  0.7363346 

L  =  79  36     4       .         .         .         sin.  9.9928075 
x  _  u  =  23   12  19  tan.  9.6321632 


Lat.  =  22  51  49       .         .         .         tan.  9.6249707 

2.  Given  the  right  ascension  of  Spica  199°  11'  35",  and  decli- 
nation 10°  19'  24"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36", 
on  the  1st  of  January,  1840,  to  find  the  longitude  and  latitude. 
Ans.  Long.  201°  36'  32",  and  lat.  2°  2'  30"  S. 


PROBLEM  XXV. 

To  compute  the  Right  Ascension  and  Declination  of  a  Heavenly 
Body  from  its  Longitude  and  Latitude^  the  Obliquity  of  the 
Ecliptic  being  given. 

The  formulae  for  the  solution  of  this  problem  are, 

log.  tang  y  —  log.  tang  X  +  ar.  co.  log.  sin  L  ; 

log.  tang' R  =  log.  cos  (y+w)  +  log.  tang  L  -f  ar.  co.  log.  cos  y— 10; 

log.  tang  D  =  log.  tang  (y  +  w)  +  log.  sin  R  —  10 ; 

in  which 

L  =  the  Longitude ; 

X  =  the  Latitude  (minus  when  South)  ; 

R  =  the  Right  Ascension  ; 

D  =  the  Declination  ; 

w  =  the  Obliquity  of  the  ecliptic  ; 

y  is  an  auxiliary  arc.     It  must  be  taken  according  to  the  sign  of 

its  tangent,  but  always  less  than  180°.    The  right  ascension  is 


364  ASTRONOMICAL  PROBLEMS. 

generally  in  the  same  quadrant  with  the  longitude.  The  declina 
tion  must  be  taken  less  than  90°,  and  will  be  north  or  south,  ac- 
cording as  the  sign  is  positive  or  negative. 

Note.  The  mean  or  apparent  obliquity  of  the  ecliptic  is  taken, 
according  as  the  given  and  required  elements  are  mean  or  apparent. 

Exam.  1.  On  the  1st  of  January,  1830,  the  longitude  of  Sirius 
was  38-  11°  44'  18",  the  latitude  39°  34'  1"  S.,  and  the  obliquity 
of  the  ecliptic  23°  27'  41"  :  required  the  right  ascension  and  de- 
clination. 

X  =  _  39°  34'    i"       .         ,         tan.  9.9171381  - 
L  =     101  44  18         .      ar.  co.  sin.  0.0091788 


y  =     139  50  14         .         .         tan.  9.9263169— 
w  =       23  27  41  


w  =  163  17  55  .  .  cos.  9.9812819  — 
L  =  101  44  18  .  .  tan.  0.6823798— 
y=  1395014  .  ar.  co.  cos.  0.1  167843— 


Right  ascen  =       99  24  48  tan.  0.7804460— 

R  =       99  24  48     '   .     '  V  I      sin.  9.9941121 
y-fw  =     163  17  55     *'  *,,?:,*         tan.  9.477  1803— 

Dec.=       16  29  20  S.    .         ,         tan.  9.  47  12924  — 

2.  Given  the  longitude  of  Aldebaran  67°  33'  5",  and  latitude 
5°  28'  38"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  36",  on  the 
1st  of  January,  1840,  to  find  the  right  ascension  and  decimation. 

Ans.  Right  ascension  66°  41  '4",  and  declination  16°  10'57"N. 


PROBLEM   XXVI. 

The  Longitude  and  Declination  of  a  Body  being  given,  and  also 
the  Obliquity  of  the  Ecliptic,  to  find  the  Angle  of  Position. 

The  formula  is 

log.  sin  p  —  log.  sin  w  -f  log.  cos  L  +  ar.  co.  log.  cos  D  —  10  : 

p  =  Angle  of  Position  (required) ; 

L  =  Longitude  ; 

D  =  Declination ; 

w  =  Obliquity  of  the  ecliptic. 

The  angle  of  position  p  must  be  taken  less  than  90°.  It  Is  to  be 
observed  also  that  when  the  longitude  is  less  than  90°,  or  more 
than  270°,  the  northern  part  of  the  circle  of  latitude  lies  to  the  west 
of  the  circle  of  declination,  but  that  when  the  longitude  is  between 
90°  and  270°,  it  lies  to  the  east. 

Note.  The  angle  of  position  may  also  be  computed  from  the 


TO  FIND  THE  TIME  OF  NEW  OR  FULL  MOON         365 

right  ascension  and  latitude,  by  means  of  a  formula  similar  to  that 
just  given,  namely, 

log.  sinp  =  log.  sin  w  +  log.  cos  R  4-  ar.  co.  log.  cos  X  —  10; 
Exarn.  1.    Given  the  longitude  of  Regulus  147°  27'  54",  and 
decimation  12°  47'  45"  N.,  and  the  obliquity  of  the  ecliptic  233 
27'  41",  to  find  the  angle  of  position. 

w  =  23°  27'  41"  .  .  sin.  9.6000260 
L=147  27  54  .  .  cos.  9.9258601 
D=  12  4745  ar.  co.cos.0.0109217 


Angle  of  pos.  =    20      7  58       .         .         sin.  9.5368078 

The  circle  of  latitude  lies  to  the  east  of  the  circle  of  declination. 
2.  Given  the  longitude  of  Fomalhaut  331°  27'  56",  and  declina- 
tion 30°  31'  14"  S.,  and  the  obliquity  of  the  ecliptic  23°  27'  41", 
to  find  the  angle  of  position.  Ans.  23°  57'  20". 

The  circle  of  latitude  lies  to  the  west  of  the  circle  of  declination 

PROBLEM  XXVII. 

To  find  from  the  Tables  the  Time  of  New  or  Full  Moon,  for  a 
given  Year  and  Month. 

For  New  Moon. 

Take  from  Table  LXXXVI,  the  time  of  mean  new  moon  in 
January,  and  the  Arguments  I,  II,  III,  and  IV,  for  the  given  year. 
Take  from  Table  LXXXVII,  as  many  lunations  with  the  corre- 
sponding variations  of  Arguments  I,  II,  III,  and  IV,  as  the  given 
month  is  months  past  January,  and  add  these  quantities  to  the  for- 
mer, rejecting  the  ten  thousands  from  the  sums  in  the  columns  of 
the  first  two  arguments,  and  the  hundreds  from  the  sums  in  the 
columns  of  the  other  two.  Seek  the  number  of  days  from  the  first 
of  January  to  the  first  of  the  given  month,  in  the  second  or  third 
column  of  Table  LXXXVIII,  according  as  the  given  year  is  a 
common  or  bissextile  year,  and  subtract  it  from  the  sum  in  the  col- 
umn of  mean  new  moon  :  the  remainder  will  be  tabular  time  of 
mean  new  moon  for  the  given  month.  It  will  sometimes  happen 
that  the  number  of  days  taken  from  Table  LXXXVIII,  will  ex- 
ceed the  number  of  days  of  the  sum  in  the  column  of  mean  new 
moon  :  in  this  case  one  lunation  more,  with  the  corresponding  ar- 
guments, must  be  added. 

With  the  sums  in  the  columns  I,  II,  III,  and  IV,  as  arguments, 
take  the  corresponding  equations  from  Table  LXXXIX,  and  add 
them  to  the  time  of  mean  new  moon  :  the  sum  will  be  the  Approxi- 
mate time  of  new  moon  for  the  given  month,  expressed  in  mean 
time  at  Greenwich. 

Next,  for  the  approximate  time  of  new  moon  calculate  the  true 
longitudes  and  hourly  motions  in  longitude  of  the  sun  and  moon  * 


366 


ASTRONOMICAL  PROBLEMS. 


subtract  the  less  longitude  from  the  greater,  and  the  houily  mo- 
tion of  the  sun  from  the  hourly  motion  of  the  moon  ;  and  say,  as 
the  difference  between  the  hourly  motions  :  the  difference  between 
the  longitudes  :  :  60  minutes  :  the  correction  of  the  approximate 
time.  The  correction  added  to  the  approximate  time,  when  the 
sun's  longitude  is  greater  than  the  moon's,  but  subtracted,  when 
it  is  less,  will  give  the  true  time  of  new  moon  required,  in  mean 
time  at  Greenwich.  This  time  may  be  reduced  to  the  meridian 
of  any  given  place  by  Problem  V. 

For  Full  Moon. 

Take  from  Table  LXXXVI,  the  time  of  mean  new  moon,  and 
the  corresponding  Arguments  I,  II,  III,  and  IV,  for  January  of  the 
given  year,  and  from  Table  LXXXVII,  a  half  lunation  with  the 
corresponding  changes  of  the  arguments.  Then,  when  the  time 
of  mean  new  moon  for  January  is  on  or  after  the  1 6th,  subtract  the 
latter  quantities  from  the  former,  increasing,  when  necessary  to 
render  the  subtraction  possible,  either  or  both  of  the  first  two  argu  - 
ments  by  10,000,  and  of  the  last  two  by  100  ;  but  add  them  when 
the  time  is  before  the  16th.  The  result  will  be  the  tabular  time 
of  mean  full  moon  and  the  corresponding  arguments,  for  January. 
Proceed  to  find  the  approximate  time  of  full  moon  after  the  same 
manner  as  directed  for  the  new  moon. 

For  the  approximate  time  of  full  moon  calculate  the  true  longi- 
tudes and  hourly  motions  in  longitude  of  the  sun  and  moon.  Sub- 
tract the  sun's  longitude  from  the  moon's,  adding  360°  to  the  latter 
if  necessary.  Take  the  difference  between  the  remainder  and  VI 
signs,  and  call  the  result  R.  Also  subtract  the  hourly  motion  of 
the  sun  from  the  hourly  motion  of  the  moon.  Then  say,  as  the 
difference  between  the  hourly  motions  :  R :  :  60m.  :  the  correction 
of  the  approximate  time.  1  he  correction  added  to  the  approxi- 
mate time  of  full  moon,  when  the  excess  of  the  moon's  longitude 
over  the  sun's  is  less  than  VI  signs,  but  subtracted  when  it  is 
greater,  will  give  the  true  time  of  full  moon. 

Exam.  1.  Required  the  time  of  new  moon  in  September,  1838 
expressed  in  mean  time  at  New  York. 


1838, 
8  lun. 

M.  New  Moon. 

I. 

II. 

m. 

IV. 

d.   h.  m. 
24  16  53 
236   5  52 

0681 
6468 

9175 
5737 

99 
22 

85 
93 

Days, 

260  22  45 
243 

7149  4912   21 
Approximate  time. 

78 

Sept'r, 

JM 

II. 
III. 
IV. 

17  22  45 
0  16 
9  35 
3 
10 

Sept'r. 

18   8  49 

TO  FIND  THE  TIME  OP  NEW  OR  FULL  MOON. 


367 


Moon's  true  long,  found  for  approx.  time,  is     5'-  25°  29'  19' 
Sun's  do.  do.  do.  5     25    27    27 


Difference, 1    52 


Moon's  hourly  motion  in  long,  is 
Sun's  do.  do. 


Difference, 27      1 

As  27'  I"  :  I'  52"  : :  60m- :  4m-  9%  the  correction. 

Approx.  time  of  new  moon,  September,         .      18d-  8h>  49™*  O8 
Correction, —  4     9 

True  time,  in  mean  time  at  Greenwich,        .      18   8    44  51 
Diff.  of  meridians, 4    56     4 

True  time,  in  mean  time  at  New  York,         .      18   3    48  47 

Exam.  2.  Required  the  time  of  full  moon  in  April,  1838,  ei- 
pressed  in  mean  time  at  New  York. 


1838, 

ilun. 

M.  Fall  Moon. 

I. 

II. 

III. 

rv. 

d.       h.       m. 
24     16     53 
14    18    22 

0681 
404 

9175 
5359 

99 

58 

85 
50 

3  lun. 

9    22    31 
88     14    12 

0277 
2425 

3816 
2151 

41 
46 

35 

97 

Days, 

98     12    43 
90 

2702 

5967 

87 

32 

April, 

II. 
III. 
IV. 

8    12    43 
8    29 
16      7 
15 
30 

Approximate  time. 

April, 

9     14      4 

Moon's  true  long,  found  for  approx.  time,  is     6'-  19°  44'  17" 


Sun's 


do. 


do. 


do. 


0     19    45    22 

5     29    58    55 
6000 


R.    . 

Moon's  hourly  motion  in  long,  is         .        • 
Sun's  do.  do. 

Difference 

As  27'  48" :  1'  5"  : :  60m-  :  2m-  20%  the  correction. 


27   48 


368  ASTRONOMICAL  PROBLEMS. 

Approximate  time  of  full  moon,  April,         .        9d*  14h    4m>  0s 
Correction, +  2  20 


True  time,  in  mean  time  at  Greenwich,       .        9    14     6  20 
Diff.  of  meridians,     .....  4  56     4 


True  time,  in  mean  time  at  New  York,       .        9      9  10  16 

3.  Required  the  time  of  new  moon  in  September,  1837,  ex- 
pressed in  mean  time  at  Philadelphia  ;  taking  the  longitudes  for 
the  approximate  time  from  the  Nautical  Almanac. 

Ans.  29d.  3h.  Om.  5s. 

4.  Required  the  time  of  full  moon,  in  October,  1837,  expressed 
in  mean  time  at  Boston.  Ans.  13d.  6h.  30m.  25s. 


PROBLEM  XXVIII. 

To  determine  the  number  of  Eclipses  of  the  Sun  and  Moon  that 
may  be  expected  to  occur  in  any  given  Year,  and  the  Times 
nearly  at  which  they  will  take  place. 

For  the  Eclipses  of  the  Sun. 

Take,  for  the  given  year,  from  Table  LXXXVI  the  time  of 
mean  new  moon  in  January,  the  arguments  and  the  number  N. 
If  the  number  N  differs  less  than  37  from  either  0,  500,  or  1000, 
an  eclipse  must  occur  at  that  new  moon.  If  the  difference  is  be- 
tween 37  and  53,  there  may  be  an  eclipse,  but  it  is  doubtful,  and 
the  doubt  can  only  be  removed  by  a  calculation  of  the  true  places 
of  the  moon  and  sun.  If  the  difference  exceeds  53,  an  eclipse  is 
impossible. 

If  an  eclipse  may  or  must  occur  at  the  new  moon  in  January, 
calculate  the  approximate  time  of  new  moon  by  Problem  XXVII, 
and  it  will  be  the  time  nearly  of  the  middle  of  the  eclipse,  express- 
ed in  mean  time  at  Greenwich.  This  may  be  reduced  to  the 
meridian  of  any  other  place  by  Problem  V. 

To  find  the  first  new  moon  after  January,  at  which  an  eclipse 
of  the  sun  may  be  expected,  seek  in  column  N  of  Table  LXXXVII 
the  first  number  after  that  answering  to  the  half  lunation,  that, 
added  to  the  number  N  for  the  given  year,  will  make  the  sum  come 
within  53  of  0,  500,  or  1000.  Take  the  corresponding  lunations, 
changes  of  the  arguments,  and  the  number  N,  and  add  them,  re- 
spectively, to  the  mean  new  moon  in  January,  the  arguments,  and 
the  number  N,  for  the  given  year.  Take  from  the  second  or  third 
column  of  Table  LXXXVIII,  according  as  the  given  year  is  a 
common  or  bissextile  year,  the  number  of  davs  next  less  than  the 
days  of  the  sum  in  the  column  of  mean  new  "moon,  and  subtract  it 
from  this  sum;  the  remainder  will  be  the  tabular  time  of  mean 
new  moon  in  the  month  corresponding  to  the  days  taken  from  Ta- 


TO  FIND  THE  NUMBER  OF  ECLIPSES  IN  A  YEAR. 


369 


ole  LXXXVIII.  At  this  new  moon  there  may  be  an  eclipse  of 
the  sun  ;  and  if  the  sum  in  the  column  N  is  within  37  of  the  num- 
bers mentioned  above,  there  must  be  one.  Find  the  approximate 
time  of  new  moon,  and  it  will  be  the  time  nearly  of  the  middle  of 
the  eclipse. 

If  any  of  the  other  numbers  in  the  last  column  of  Table 
LXXX  V II  are  found,  when  added  to  the  number  N  of  the  given 
year,  to  give  a  sum  that  falls  within  the  limit  53,  proceed  in  a  simi- 
lar manner  to  find  the  approximate  times  of  the  eclipses. 

Note.  When  the  sum  of  the  numbers  N,  or  the  number  N  itself, 
in  case  the  eclipse  happens  in  January,  is  a  little  above  0,  or  a 
little  less  than  500,  the  moon  will  be  to  the  north  of  the  sun,  and 
there  is  a  probability  that  the  eclipse  will  be  visible  at  any  given 
place  in  north  latitude  at  which  the  approximate  time  of  the  eclipse, 
found  as  just  explained  and  reduced  to  the  meridian  of  the  place, 
comes  during  the  day-time.  When  the  number  N  found  for  the 
eclipse  is  more  than  500,  the  moon  will  be  to  the  south  of  the  sun, 
and  the  eclipse  will  seldom  be  visible  in  the  northern  hemisphere, 
except  near  the  equator. 

For  the  Eclipses  of  the  Moon. 

Find  the  time  of  full  moon  and  the  corresponding  arguments  and 
number  N,  for  January  of  the  given  year,  as  explained  in  Problem 
XXVII.  Then  proceed  to  find  the  times  at  which  eclipses  of  the 
moon  may  or  must  occur,  after  the  same  manner  as  for  eclipses  of 
the  sun,  only  making  use  of  the  limits  35  and  25,  instead  of  53 
and  37.* 

Note.  An  eclipse  of  the  moon  will  be  visible  at  a  given  place, 
if  the  time  of  the  eclipse  thus  found  nearly,  and  reduced  to  the 
meridian  of  the  place,  comes  in  the  night. 

Exam.  1.  Required  the  eclipses  that  maybe  expected  in  the 
year  1840,  and  the  times  nearly  at  which  they  will  take  place. 

For  the  Eclipses  of  the  Sun. 


1840, 
2  lun. 

M.  New  Moon. 

I. 

II. 

m. 

IV. 

N. 

d.       h.       m. 
3     10    30 
59      1    28 

0085 
1617 

6386 
1434 

65 
31 

63 

98 

844 
170 

62    11    58 
60 

1702     7820       96         61        014 

As  the  sum  of  the  numbers  N 
comes  within  37  of  0,  there  must  be 
an  eclipse. 

Mean  time  at  Greenwich. 

March, 
I. 
II. 
III. 
IV. 

2    11     58 
8      3 
19     38 
12 
13 

March, 

3    16      4 

*  The  numbers  53,  37,  and  35,  25,  are  the  lunar  and  solar  ecliptic  limits,  as 
determined  by  Delambre.  The  limits  given  in  the  text,  converted  into  thousandth 
•arts  of  the  circle,  are  55,  37,  and  37,  21. 

24 


370 


ASTRONOMICAL  PROBLEMS. 


1840, 
8  lun. 

M.  New  Moon. 

I. 

II. 

III. 

IV. 

N. 

d.        h.      m. 
3      10     30 

236      5    52 

0085 
6468 

6386 
5737 

65 
22 

63 
93 

844 
682 

239     16    22 
213 

6553     2123       87         56        526 

As  the  sum  of  the  numbers  N 
comes  within  37  of  500,  there  must 
be  an  eclipse. 

Mean  time  at  Greenwich 

August, 

II*. 
III. 
IV. 

26     16    22 
0    54 
0    49 
15 
16 

August, 

26    18    36 

For  the  Eclipses  of  the  Moon. 


1840, 
ilun. 

M.  Full  Moon. 

I. 

II. 

m. 

IV. 

N. 

d.       h.       m. 
3      10     30 
14    18    22 

0085 
404 

6386 
5359 

65 

58 

63 
50 

844 
43 

llun. 

18      4    52 
29     12    44 

489 
808 

1745 
717 

23 
15 

13 

99 

887 
85 

47     17    36 
31 

1297     2462       38         12        972 

As  the  sum  of  the  numbers  N,  al- 
though it  comes  within  35  of  1000, 
does  not  come  within  25,  the  eclipse 
may  be  considered  doubtful. 

Mean  time  at  Greenwich. 

Febr. 
I. 
II. 
III. 
IV. 

16    17    36 
7    27 
0    23 
5 
27 

Febr. 

17      1    58 

1840, 
7  lun. 

M.  Full  Moon. 

I. 

H. 

III. 

IV. 

N. 

d.       h.       m. 
18      4    52 
206    17      8 

489 
5659 

1745 
5020 

23 

7 

13 
94 

887 
596 

224    22      0 
213 

6148  1  6765       30         07        483 

As  the  sum  of  the  numbers   N 
comes  within  25  of  500,  there  moat 
be  an  eclipse. 

Mean  time  at  Greenwich. 

August, 

II.' 
III. 
IV. 

11    22      0 
1    37 
19    16 
3 
25 

August, 

12    19    21 

2.  Required  the  eclipses  that  may  be  expected  in  the  year  1839, 
and  the  times  nearly  at  which  they  will  take  place,  expressed  in 
mean  civil  time  at  New  York. 


TO  CALCULATE  A  LUNAR  ECLIPSE.  371 

Ans.  One  of  the  sun  on  the  15th  of  March,  at  9h.  20m  A.  M. ; 
and  one  of  the  sun  on  the  7th  of  September,  at  5h.  24m.  P.  M. 

3.  Required  the  eclipses  that  may  be  expected  in  the  year  1841 
and  the  times  nearly  at  which  they  will  take  place,  expressed  ir. 
mean  civil  time  at  New  York. 

Ans.  Four  of  the  sun,  namely,  one  on  the  22d  of  January,  Pt 
12h.  18m.  P.  M. ;  one  on  the  21st  of  February,  at6h.  17m.  A.M  ; 
one  on  the  18th  of  July,  at  9h.  24m.  A.  M. ;  and  one  on  the  16m 
of  August,  at  4h.  28m.  P.  M. :  and  two  of  the  moon,  namely,  one 
on  the  5th  of  February,  at  9h.  10m.  P.  M. ;  and  one  on  the  2d  of 
August,  at  5h.  5m.  A.  M. 

The  eclipses  of  the  sun  in  January  and  August  may  be  con- 
sidered as  doubtful. 


PROBLEM   XXIX. 
To  calculate  an  Eclipse  of  the  Moon. 

The  calculation  of  the  circumstances  of  a  lunar  eclipse  is  effect- 
ed with  the  following  fundamental  data,  derived  from  the  tables  of 
the  sun  and  moon : 

Approximate  Time  of  Full  Moon  (at  Greenwich),  T 

Sun's  Longitude  at  that  time,  L 

Do.  Hourly  Motion,  ....  s 

Do.  Semi-diameter, S 

Do.  Parallax, p 

Moon's  Longitude, / 

Do.      Latitude,       ......  X 

Do.      Equatorial  Parallax,  *         .  P 

Do.      Semi-diameter,      .         .         .         .         .  d 

Do.      Hourly  Motion  in  longitude,  m 

Do.      Hourly  Motion  in  latitude,     .        .  n 

We  obtain  the  time  T  by  Problem  XXVII ;  the  quantities  ap- 
pertaining to  the  sun,  namely,  L,  s,  and  £,  by  Problem  IX  ;*  and 
those  which  have  relation  to  the  moon,  namely,  Z,  X,  P,  d,  m,  and 
n,  by  Problem  XIV. 

From  these  quantities  we  derive  the  following : 

True  Time  of  Full  Moon,  (at  given  place,)         .  T' 

Moon's  Latitude  at  that  time,    ....  X' 

Semi-diameter  of  earth's  shadow,  S 

Inclination  of  Moon's  relative  orbit,    ...  I 

T  being  known,  T'  is  found  as  explained  in  Problem  XXVII. 
To  obtain  X',  we  state  the  following  proportion, 

1  hour  :  correction  for  the  time  of  full  moon  :  :  n :  x ; 

*  p  may  be  taken  as  9". 


372  ASTRONOMICAL  PROBLEMS. 

from  this  we  deduce  the  value  of  x  ;  and  thence  find  X  by  the 
equation 

X'  =  X  ±  x. 

When  the  true  time  of  full  moon,  expressed  in  mean  cime  ai 
Greenwich,  is  later  than  the  approximate  time,  the  upper  sign  is 
to  be  used,  if  the  latitude  is  increasing,  the  lower  if  it  is  decreas- 
ing ;  but  when  the  true  time  is  earlier  than  the  approximate  time, 
the  lower  sign  is  to  be  used  if  the  latitude  is  increasing  ;  the  upper 
if  it  is  decreasing. 

The  value  of  S  is  derived  from  the  equation 


and  the  angle  I  from  the  formula 

log.  tang  I  =  log.  n  +  ar.  co.  log.  (m  —  s). 
The  foregoing  quantities  having  all  been  determined,  the  various 
circumstances  of  the  eclipse  may  be  calculated  by  the  following 
formulae  : 

For  the  Time  of  the  Middle  of  the  Eclipse. 

3.55630  +  log.  cos  I  -f-  ar.  co.  log.  (m  —  s)  —  20  =  R  ; 

log.  t  =  R  +  log.  X'  +  log.  sin  I  —  10  ; 

M=T'±£: 

t  =  interval  between  time  of  middle  of  eclipse  and  time  of  full 
moon  ;  M  =  time  of  middle  of  the  eclipse. 

The  upper  sign  is  to  be  taken  in  the  last  equation  when  the  lati- 
tude is  decreasing;  the  lower,  when  it  is  increasing. 

For  the  Times  of  Beginning  and  End. 

log.  c  =  log  X'  +  log.  cos  I  —  10  ; 
log<  v  _  %  (S  .+  d  +  c)  +  log.  (S  4-  d-  c)  {  R  . 

£  =  M  —  v,  and  E  =  M  +  v  : 

v  =  half  duration  of  the  eclipse  ;  B  =  time  of  beginning  ;  and  E  = 
time  of  end. 

Note.  If  c  is  equal  to  or  greater  than  S  +  d,  there  cannot  be  an 
eclipse. 

For  the  Times  of  Beginning  and  End  of  the  Total  Eclipse. 


log.  v>  =      .  .  +  R  . 

B'  =  M  —  v1,  and  E;  =  M  +  v'  : 

v'  =  half  duration  of  the  total  eclipse  ;  B'  =  time  of  beginning  of 
total  eclipse  ;  and  E;  =  time  of  end  of  total  eclipse. 

Note.  When  c  is  greater  than  S  —  d,  the  eclipse  cannot  be  total. 

For  the  Quantity  of  the  Eclipse. 

log.  Q  =  0.77815  +  log.  (S  +  d  -  c)  +  ar.  co.  log.  d  —  10  : 
Q  =  the  quantity  of  the  eclipse  in  digits. 


TO  CALCULATE  A  LUNAR  ECLIPSE.  373 

Note  1 .  An  eclipse  of  the  moon  begins  on  the  eastern  limb,  and 
ends  on  the  western.  In  partial  eclipses  the  southern  part  of  the 
moon  is  eclipsed  when  the  latitude  is  north,  and  the  northern  part 
when  the  latitude  is  south. 

Note  2.  When  the  eclipse  commences  before  sunset,  and  ends 
after  sunset,  the  moon  will  rise  more  or  less  eclipsed.  To  obtain 
the  quantity  of  the  eclipse  at  the  time  of  the  moon's  rising,  find 
the  moon's  hourly  motion  on  the  relative  orbit  by  the  equation 

log.  h  =  log.  (m —  s)  +  ar.  co.  log.  cos  I ; 

in  which  h  =  hourly  motion  on  relative  orbit.  Also  find  the  inter 
val  between  the  time  of  sunset  and  the  time  of  the  middle  of  the 
eclipse,  which  call  i.  Then, 

1  hour  :  i  :  :  h  :  x. 

Deduce  the  value  of  x  from  this  proportion,  and  substitute  it  in 
he  equation  

c  ^v/^  +  z2; 

in  which  c  designates  the  same  quantity  as  in  previous  formulae. 
Find  the  value  of  c',  and  use  it  in  place  of  c  in  the  above  formula 
for  the  quantity  of  the  eclipse,  and  it  will  give  the  quantity  of  the 
eclipse  at  the  time  of  the  moon's  rising.  When  the  eclipse  begins 
before  and  ends  after  sunrise,  the  quantity  of  the  eclipse  at  the 
time  of  the  moon's  setting  may  be  found  in  the  same  manner,  only 
using  sunrise  instead  of  sunset. 

Example.  Required  to  calculate,  for  the  meridian  of  New  York, 
the  eclipse  of  the  moon  in  October,  1837. 


Elements. 


Approximate  time  of  full  moon 
Sun's  longitude  at  that  time, 

Do.  hourly  motion, 

Do.  semi-diameter, 

Do.  parallax, 
Moon's  longitude,    . 

Do.  latitude,          .         . 

Do.  equatorial  parallax, 

Do.  semi-diameter, 

Do.  hourly  motion  in  long. 


T  =  llh-  10m-(0ct.  13) 

L  =  6*  20°  24'  28" 
s   =  2  29 

<5   =  16     4 

=  9 

=  0    20   21   51 
X  =  11   28  S. 

P  =  59  32 

d  =  16   13 

m=  35  54 


Do.  hourly  motion  in  lat.  (tending  north),  n  =  3   19 


Approx.  time  of  full  moon,  October,          .  13d*  llh*  10™* 

Correction  found  by  Prob.  XXVII,           .  +  4     42 

True  time,  in  mean  time  at  Greenwich,    .  13    11    14    42 

DifF.  of  meridians,  .....  4    56      4 


True  time,  in  mean  time  at  New  York,  T  =   13     6    18     38 


ASTRONOMICAL   PROBLEMS. 

Moon's  lat.  at  approx.  time,           .         .         .        X  =  11'  28"  Sa 
Correction, a?  =     — 16 

Moon's  lat.  at  true  time,       .        .        .        .        X'=  11    12 

Moon's  equatorial  parallax, P  =  59'  32' 

Sun's  do  p  =         9 

Sum, t  .        .  59  41 

Sun's  semi-diameter,   .        .        .     ,  ^;;;  -f       ^    5  =  16    4 

Diff. P+p— -5=4337 

Add  YV(P-l~p —  <$)  =       44 

Semi-diameter  of  earth's  shadow,  .         .  .      .     S  =  44  21 

Moon's  hor.  mot.  less  sun's  (m  —  s)  =  2005"  .  ar.  co.  log.  6.69789 
Moon's  hor.  motion  in  latitude,     n  =    199    .        .    log.  2.29885 

Inclination  of  rel.  orbit,  I  =  5°  40'         .         .         .    tan.  8.99674 

Time  of  Middle. 

3.55630 

1    .    ,    .    .    .    5°40'  .    .    cos.  9.99787 
01  —  s    .    .    .    .        2005"  ar.  co.  log.  6.69789 

R.  0.25206 

X'        .         .  .         .  672"         .    log.  2.82737 

I         .     ...         .         .         .         5°  40'     ...    sin.  8.99450 

t  &•    lm-  58s-  =  118s-     .     .    log.  2.07393 

T'       .  6    18     38P.M. 

Middle,        .  6    20     36P.M. 

Times  of  Beginning  and  End. 

X' log.  2.82737 

I  .         ,    cos.  9.99787 


11' 9"  =  669" 

4303" 
2965 


46*-  22'-  =  63821- 


log.  2.82524 

log.  3.63377 
log.  3.47202 

2  )Tl0579 

3.55289 
R.  0.25206 

log.  3.80495 


Middle,       ! 

Beginning, . 
End, 


S-J-fc 
S-d-c 


Middle, 


TO  CALCULATE  A  SOLAR  ECLIPSE 

lh-  46m-  22s-  =  63821- 
6  20  36 


4  34  14  P.  M. 
8   6  58P.M. 


2357" 
1019 


Qh.  46m.  9..=  2769* 
6  20  36 


Beg.  of  total  eclipse,    5    34     27  P.  M. 
End  of  total  eclipse,    7      6     45  P.  M 


S+d-c 
d 

Quantity, 


18.3  digits, 


log.  3.80495 


log.  3.37230 
log.  3.00817 

2  )  6.38053 

3.19026 
R     0.25206 

log.  3.44232 


0.77815 

log.  3.47202 

973".      ar.  co.  log.  7.01189 

log.  1.26200 


PROBLEM  XXX. 
To  calculate  an  Eclipse  of  the  Sun,  for  a  given  Place. 

Having  found  by  the  rule  given  in  the  note  to  Problem  XXVIII, 
that  there  is  a  probability  that  the  eclipse  will  be  visible  at  the 
given  place,  and  calculated  the  approximate  time  of  new  moon  by 
Problem  XXVII,  find  from  the  tables,  for  this  time  or  for  the  near- 
est whole  or  half  hour,  the  sun's  longitude,  hourly  motion,  and 
semi-diameter  ;  and  the  moon's  longitude,  latitude,  equatorial  par- 
allax, semi-diameter,  and  hourly  motions  in  longitude  and  latitude. 
Find  also  by  Problem  XVI,  the  longitude  and  altitude  of  the 
ncnagesimal  degree  ;  and  thence  compute  by  Problem  XVII,  the 
apparent  longitude,  latitude,  and  augmented  semi-diameter  of  the 
moon,  (using  the  relative  horizontal  parallax.)  With  these  data 
compute  the  apparent  distance  of  the  centres  of  the  sun  and  moon,, 
at  the  time  in  question,  by  means  of  the  following  formulae : 

log.  tang  &  =  log.  X'  +  ar.  co.  log.  a  ; 
log.  A  =  log.  a  +  ar.  co.  log.  cos  d  : 


376  ASTRONOMICAL  PROBLEMS. 

in  which 

A  =  appar.  distance  of  centres  ; 
X'  ==  appar.  Lat.  of  Moon  ; 

a   =  Diff.  of  appar.  Long,  of  Moon  and  Sun  =  diff.  of  f.ppar 
long,  of  Moon  (found  as  above)  and  true  long,  of  Sua 

6  is  an  auxiliary  arc.  The  value  of  6  being  derived  from  the 
first  equation,  the  second  will  then  make  known  the  value  of  A. 

a  and  X;  are  in  every  instance  to  be  affected  with  the  positive 
sign.* 

For  the  Approximate  Times  of  Beginning,  Greatest  Obscuration, 

and  End. 

Let  the  time  for  which  the  above  calculations  are  made,  be  de- 
noted by  T.  If  the  apparent  distance  of  the  centres  of  the  sun 
and  moon,  found  for  the  time  T,  is  less  than  the  sum  of  their  ap- 
parent semi-diameters,  there  is  an  eclipse  at  this  time.  But  if  ii 
is  greater,  either  the  eclipse  has  not  yet  commenced,  or  it  has  al- 
ready terminated.  It  has  not  commenced  if  the  apparent  longitude 
of  the  moon  is  less  than  the  longitude  of  the  sun  ;  and  has  termi- 
nated, if  the  apparent  longitude  of  the  moon  is  greater  than  the 
longitude  of  the  sun. 

1.  If  there  should  be  an  eclipse  at  the  time  T,  from  the  sun's 
longitude  and  hourly  motion  in  longitude,  and  the  moon's  longi- 
tude and  latitude,  and  hourly  motions  in  longitude  and  latitude, 
found  for  this  time,  calculate  the  longitudes  and  the  moon's  lati- 
tude for  two  instants  respectively  an  hour  before,  and  an  hour  after 
the  time  T.  The  semi-diameter  of  the  sun,  and  the  equatorial 
parallax  and  semi-diameter  of  the  moon,  may,  in  our  present  in- 
quiry, be  regarded  as  remaining  the  same  during  the  eclipse.  Find 
the  apparent  longitude  and  latitude,  and  the  augmented  semi-diam- 
eter of  the  moon,  (using  in  all  cases  the  relative  parallax,)  and 
thence  compute  by  the  formulae  already  given,  the  apparent  dis 
tance  of  the  centres  of  the  sun  and  moon  at  the  two  instants  in 
question. 

Observe  for  each  result,  whether  it  is  less  or  greater  than  the 
sum  of  the  apparent  semi-diameters  of  the  two  bodies.  If  the 
moon  is  apparently  on  the  same  side  of  the  sun  at  the  times  T  and 
T  -f  lh.,  take  the  difference  of  the  distances  of  the  two  bodies  in 
apparent  longitude  at  these  times,  but,  if  it  is  on  opposite  sides, 
take  their  sum,  and  it  will  be  the  variation  of  this  distance  in  the 

*  A,  the  apparent  distance  of  the  centres,  may  be  found  without  the  aid  of  loga- 
rithms by  means  of  the  following  equation  : 


A-2. 

If  the  logarithmic  formulae  are  used,  it  will  be  sufficient  here  to  take  out  the  angle 
0  to  the  nearest  minute.  When  we  have  occasion  to  obtain  the  distance  of  the 
centres  exact  to  within  a  small  fraction  of  a  second,  6  must  be  taken  to  the  nearest 
tens  of  seconds,  if  it  exceeds  20°  or  30°. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  377 

hour  following  T.  Find  in  like  manner  the  variation  of  the  dis- 
tance during  the  hour  preceding  T.  Then,  if  the  apparent  distance 
of  the  centres  at  the  times  (T  —  lh.),  (T  +  Ih.)  is  less  than  the 
sum  of  the  apparent  semi-diameters,  deduce  from  these  results 
the  variations  of  the  distance  in  apparent  longitude  during  the  pre- 
ceding and  following  hours,  allowing  for  the  second  difference,  and 
observing  whether  the  two  bodies  are  approaching  each  other,  or 
receding  from  each  other.  Thence,  find  the  distance  in  apparent 
longitude  at  the  times  (T  —  2h.),  (T  +  2h.)  Find  by  the  same 
method  the  apparent  latitude  of  the  moon  at  the  instants  (T  —  2h.), 
(T  -f  2h.),  observing  that  the  variation  of  the  apparent  latitude  in 
any  given  interval  is  the  difference  between  the  latitudes  at  the 
beginning  and  end  of  it,  if  they  are  both  of  the  same  name ;  their 
sum,  if  they  are  of  opposite  names. 

From  these  results  derive  the  apparent  distance  of  the  centres 
of  the  sun  and  moon  at  the  two  instants  in  question. 

If  there  should  still  be  an  eclipse  at  the  time  (T  +  2h.)  or 
(T  —  2h.),  find  by  the  same  method  the  distance  of  the  centres  at 
the  time  (T  +  3h.)  or  (T  -  3h.)  These  calculations  being  effect- 
ed, the  times  of  the  beginning,  greatest  obscuration,  and  end  of  the 
eclipse,  will  fall  between  some  of  the  instants  T,(T-  lh.),(T+  lh.), 
&c.,  for  which  the  apparent  distance  of  the  centres  is  computed. 

2.  If  the  eclipse  occurs  after  the  time  T,  the  different  phases 
will  happen  between  the  instants  T,  (T  +  lh.),  (T  +  2h.),  &c. 
Find  the  apparent  distance  of  the  centres  of  the  sun  and  moon  for 
the  times  (T  +  lh.),  (T  +  2h.),  by  the  same  method  as  that  by 
which  it  is  found  for  the  times  (T  +  lh.),  (T  —  lh.),  in  the  case 
just  considered.     Then,  if  the  eclipse  has  not  terminated,  deduce 
the  distance  of  the  moon  from  the  sun  in  apparent  longitude,  and 
the  moon's  apparent  latitude,  for  the  time  (T  +  3h.),  from  these 
distances  and  latitudes  at  the  times  T,  (T  -f  Ih.),  (T  -f  2h.) ;  as 
in  the   preceding   case  the   distance   and  latitude  for  the   time 
(T-f2h.)  were  deduced  from  the  same  at  the  times  (T  —  lh.),  T, 
(T  +  lh.)     With  the  results  obtained  compute  the  apparent  dis- 
tance of  the  centres  of  the  two  bodies  at  the  time  (T  4-  3h.) 

3.  In  case  the  eclipse  occurs  before  the  time  T,  the  apparent 
distance  of  the  centres  must  be  found  by  similar  methods  for  the 
times  (T  -  lh.),  (T  -  2h.),  &c. 

The  calculation  is  to  be  continued  until  the  distance,  from  being 
less,  becomes  greater  than  the  sum  of  the  semi-diameters. 

Now,  let  h  —  variation  of  apparent  distance  of  centres  in  the 
interval  of  one  hour  comprised  between  the  first  two  of  the  instants 
for  which  the  distance  is  computed ;  d  =  difference  between  the 
sum  of  the  semi-diameters  of  the  sun  and  moon  and  the  apparent 
distance  of  their  centres  at  the  first  instant ;  and  t  =  interval  be- 
tween first  instant  and  the  time  of  the  beginning  of  the  eclipse 
Then, 

h:d::  60m-  •  t  (nearly.) 


378  ASTRONOMICAL  PROBLEMS. 

Find  the  value  of  t  given  by  this  proportion,  and  add  it  to  the 
time  at  the  first  instant,  and  the  result  will  be  a  first  approximation 
to  the  time  of  the  beginning  of  the  eclipse,  which  call  b.  Find, 
by  interpolation,*  the  distance  of  the  moon  from  the  sun  in  appa 
rent  longitude  (a),  and  the  moon's  apparent  latitude  (X'),  for  this 
time,  and  thence  compute  the  apparent  distance  of  the  centres. 
Take  h  =  variation  of  apparent  distance  in  the  interval  between  the 
time  b  and  the  nearest  of  the  two  instants  above  mentioned,  ber 
tween  which  the  beginning  falls,  and  d  =  difference  between  the 
apparent  distance  of  the  centres  at  the  time  b  and  the  sum  of  the 
semi-diameters,  and  compute  again  the  value  of  t.  Add  this  to  the 
time  6,  or  subtract  it  from  it,  according  as  b  is  before  or  after  the 
beginning,  and  the  result  will  be  a  second  approximation  to  the 
time  of  the  beginning,  which  call  B.  A  result  still  more  approxi- 
mate may  be  had,  by  taking  h  =  variation  of  apparent  distance  of 
centres  in  the  interval  B  —  b,  d  =  difference  between  apparent  dis- 
tance at  the  time  B  and  sum  of  semi-diameters,  finding  anew  the 
value  of  t  given  by  the  preceding  proportion,  and  adding  it  to,  or 
subtracting  it  from,  as  the  case  may  be,  the  time  B.  But  prepara 
tory  to  the  calculation  of  the  exact  times,  it  will  suffice,  in  general, 
to  take  the  first  approximation. 

The  end  of  the  eclipse  will  fall  between  the  last  two  of  the 
several  instants  for  which  the  apparent  distance  of  the  centres  of 
the  moon  and  sun  have  been  computed.  The  approximate  time 
of  the  end  is  found  by  the  same  method  as  that  of  the  beginning.! 

*  The  second  differences  may  easily  be  taken  into  the  account  in  finding  the 
quantities  a  and  A'  for  the  time  b.  Thus,  let  k  =  variation  of  a  for  the  interval  of 
an  hour  comprised  between  the  instants  above  mentioned,  k  =  same  for  the  suc- 
ceeding hour,  and  i  =  interval  between  b  and  the  nearer  of  the  two  instants,  (in 

It  if ffi 

minutes.)     Then,  if  we  put  /=  — ,  c  =  — — — ,  and  t>  =  var.  of  a  in  interval  i, 

b  ob 


10 

The  upper  sign  is  to  be  used  when  the  time  b  is  nearer  the  first  than  the  second 
instant,  the  lower  when  it  is  nearer  the  second  than  the  first,  c  is  to  be  used  with 
its  sign.  The  error  by  this  method  will  not  exceed  the  number  c,  (supposing  the 
changes  of  A-,  k',  from  10m.  to  10m.  to  increase  or  decrease  by  equal  degrees.) 

The    general    formula    for    interpolation    is    Q  =q  +  -d'-\  --    ~      d"  + 

-  o~^^2  -  d"f+  &:.,  in  which  q  is  the  first  of  a  series  of  values,  found  at 
A  .  o  .  n* 

equal  intervals,  of  the  quantity  whose  value  Q  at  the  time  t  is  sought,  t  is  reck- 
oned from  the  time  for  which  q  is  found,  h  is  one  of  the  equal  intervals,  d',  d"f 
d"',  &c.,  are  the  first,  second,  third,  &c.,  differences.  If  we  make  h  =  1,  we  have 


t  In  effectiag  the  reductions  of  the  quantities  a  and  A'  to  the  first  approximate 
time  of  end,  k  must  stand  for  the  variation  of  a  during  the  hour  preceding  that 
comprised  between  the  last  two  instants,  and  the  last  instant  must  be  substituted 
for  the  first.  (See  Note  above.) 


TO  CA.CULATE  A  SOLAR  ECLIPSE. 


379 


The  middle  of  the  interval  between  the  approximate  times  of 
the  beginning  and  end  of  the  eclipse,  will  be  a  first  approximation 
to  the  time  of  greatest  obscuration. 

Note.  When  the  object  is  merely  to  prepare  for  an  ooservation 
results  sufficiently  near  the  truth  may  be  obtained  by  a  graphical 
construction.  The  elements  of  the  construction  are  the  difference 
of  the  apparent  longitudes  of  the  moon  and  sun,  and  the  apparent 
latitude  of  the  moon,  found  as  above,  for  two  or  more  instants  du- 
ring the  continuance  of  the  eclipse.  Draw  a  right  line  EF,  (Fig. 
119,)  to  represent  the  ecliptic,  assume  on  it  some  point  C  for  the 

Fig.  119. 


position  of  the  sun  at  the  instant  of  apparent  conjunction,  and  lay 
off  CA,  CA',  equal  to  the  two  differences  of  apparent  longitude  ; 
and  to  the  right  or  left,  according  as  the  moon  is  to  the  west  or 
east  of  the  sun  at  the  instants  for  which  the  calculations  have  been 
made.  Erect  the  perpendiculars  Ap,  A'p',  and  mark  off  Aa,  AW 
equal  to  the  two  apparent  latitudes.  Through  a,  a',  draw  a  right 
line,  and  it  will  be  the  apparent  relative  orbit  of  the  moon,  or 
will  differ  but  little  from  it.  From  C  let  fall  the  perpendicular  Cm 
upon  the  relative  orbit,  m  will  be  the  apparent  place  of  the  moon 
at  the  instant  of  greatest  obscuration.  Take  a  distance  in  the  di- 
viders equal  to  the  sum  of  the  apparent  semi-diameters  of  the  moon 
and  sun,  and  placing  one  foot  of  it  at  C,  mark  off  with  the  other 
the  points  /,  /',  for  the  beginning  and  end  of  the  eclipse,  and  by 
means  of  a  square  mark  on  EF  the  points  b,  e,  which  answer  to 
the  beginning  and  end.  If  the  eclipse  be  total  or  annular,  mark 
the  points  of  immersion  and  emersion,  g,  g',  with  an  opening  in 
the  dividers  equal  to  the  difference  of  the  semi-diameters,  and  find 
the  corresponding  points  &',  e'  on  the  line  EF. 

If  the  calculations  are  made  from  hour  to  hour,  the  distance  AA' 
is  the  apparent  relative  hourly  motion  of  the  sun  and  moon  in  lon- 
gitude. This  distance  laid  off  repeatedly  to  the  right  and  left  will 
determine  the  points  1,  2,  &c.,  answering  to  lh.,  2h.,  &c.  before 


380  ASTRONOMICAL  PROBLEMS. 

and  after  the  times  for  which  the  calculations  are  made.  If  the 
spaces  in  which  the  points  bt  e,  answering  to  the  beginning  and 
end  of  the  eclipse,  occur,  be  divided  into  quarters,  and  then  sub- 
divided into  three  equal  parts  or  five-minute  spaces,  the  approxi- 
mate times  of  the  beginning  and  end  of  the  eclipse  will  become 
known. 

From  the  point  m,  as  a  centre,  describe  the  lunar  disc ;  and 
from  the  point  C,  as  a  centre,  describe  the  sun's  disc,  and  we  shall 
have  the  figure  of  the  greatest  eclipse.  The  quantity  of  the  eclipse 
will  result  from  the  proportion 

SN  :  MN  : :  12  :  number  of  digits  eclipsed. 

Draw  from  the  centre  C  to  the  place  of  commencement/,  the 
line  C/;  and  through  the  same  point  C  raise  a  perpendicular  to 
the  ecliptic.  With  the  longitude  of  the  sun  at  the  time  of  the  be- 
ginning, calculate  its  angle  of  position  by  Problem  XIII,  and  lay  it 
off  in  the  figure,  placing  the  circle  of  declination  CP  to  the  left  if 
the  tangent  of  the  angle  of  position  be  positive,  to  the  right  if  it  be 
negative. 

Compute  also  for  the  time  of  beginning  the  angle  of  the  vertical 
circle  of  the  sun  with  the  circle  of  declination,  that  is,  the  angle 
PSZ  in  Fig.  6  p.  13,  for  which  we  have  in  the  triangle  PSZ 
the  side  PS  =  co-declination,  the  side  PZ  =  co-latitude,  and  the 
included  angle  ZPS.  (The  requisite  formulae  are  given  in  the  Ap- 
pendix.) Form  this  angle  in  the  figure  at  the  point  C,  placing  CZ 
to  the  right  or  left  of  CP,  according  as  the  time  is  in  the  forenoon 
or  afternoon ;  CZ  will  be  the  vertical,  and  Z  the  vertex,  or  highest 
point  of  the  sun.  The  arc  Tit  on  the  limb  of  the  sun  will  be  the 
angular  distance  from  the  vertex  of  the  point  on  the  limb  at  which 
the  eclipse  commences. 

For  the  True  Times  of  Beginning,  Greatest  Obscuration,  and  End. 
The  approximate  times  of  beginning,  greatest  obscuration,  and 
end  of  the  eclipse,  being  calculated  by  the  rules  which  have  been 
given,  find  from  the  tables,  or  from  the  Nautical  Almanac,  (see 
Problem  XXXI,)  the  moon's  longitude,  latitude,  equatorial  paral- 
lax, semi-diameter,  and  hourly  motions  in  longitude  and  latitude,  for 
the  approximate  time  of  greatest  obscuration.*  With  the  moon's 
longitude  and  latitude,  and  hourly  motions  in  longitude  and  latitude, 
found  for  this  time,  calculate  the  longitude  and  latitude  for  the  ap- 
proximate times  of  beginning  and  end.  The  parallax  and  semi- 
diameter  may,  without  material  error,  be  considered  the  same 
during  the  eclipse.  With  the  moon's  true  longitude,  latitude,  and 
semi-diameter  at  the  approximate  times  of  beginning,  greatest  ob- 
scuration, and  end,  calculate  its  apparent  longitude  and  latitude, 

*  It  will,  in  general,  suffice  to  calculate  the  moon's  longitude  and  latitude  from 
the  elements  already  found  for  the  approximate  time  of  full  moon,  if  these  have 
been  accurately  determined.  The  equatorial  parallax  and  semi-diameter  may  be 
found  by  interpolation  from  the  Nautical  Almanac. 


TO  CALCULATE  A  SCLAR  ECLIPSE.  381 

and  augmented  semi-diameter,  for  these  several  times,  (making  use 
of  the  relative  parallax.)  With  the  sun's  longitude  and  hourly  mo- 
tion previously  found  for  the  approximate  time  of  new  moon,  find 
his  longitude  at  the  approximate  times  of  beginning,  greatest  ob- 
scuration, and  end.  The  sun's  semi-diameter  found  for  the  ap- 
proximate time  of  new  moon,  will  serve  also  for  any  time  during 
the  eclipse.  With  the  data  thus  obtained,  calculate  by  the  formu- 
lae given  on  page  375  the  apparent  distance  of  the  centres  of  the 
sun  and  moon  at  the  approximate  times  of  the  three  phases. 

Note.  When  very  great  accuracy  is  required,  the  moon's  longi- 
tude, latitude,  equatorial  parallax,  semi-diameter,  and  hourly  mo- 
tions in  longitude  and  latitude,  must  be  calculated  directly  from 
the  tables,  or  from  the  Nautical  Almanac,  for  the  approximate 
times  of  the  beginning  and  end,  as  well  as  for  that  of  the  greatest 
obscuration. 

For  the  Beginning. 

Subtract  the  apparent  longitude  of  the  moon  at  the  approximate 
time  of  beginning  from  the  true  longitude  of  the  sun  at  the  same 
time,  and  denote  the  difference  by  a.  Do  the  same  for  the  approx- 
imate time  of  greatest  obscuration.  Subtract  the  latter  result  from 
the  former,  paying  attention  to  the  signs,  and  call  the  remainder  k. 
Next,  take  the  difference  between  the  apparent  latitudes  of  the 
moon  at  the  approximate  times  of  beginning  and  greatest  obscura- 
tion, if  they  are  of  the  same  name  ;  their  sum,  if  they  are  of  oppo- 
site names  ;  and  denote  the  difference  or  sum,  as  the  case  may  be, 
by  n.  This  done,  compute  the  correction  to  be  applied  to  the  ap- 
proximate time  of  beginning  by  means  of  the  following  formulae  : 
log.  b  =  log.  a  -f-  log.  k  4-  ar.  co.  log.  n  —  10  ; 


log  t  =  log.  (S  +  A)  -f  log.  (S  -  A)  +  ar.  co.  log.  n  +  ar, 

co.  log.  c  -f-  log.  L  +  1.47712  —  20  : 
in  which 

t  =  Correction  of  approx.  time  of  beginn.  (required)  ; 

a  =  Diff.  of  appar.  long,  of  Moon  and  Sun  at  approx.  time  ; 

L=  Half  duration  of  eclipse  in  minutes  (known  approximately)  ; 

k  =  Appar.  relative  motion  of  Sun  and  Moon  in  long,  in  the  in- 
terval L  ; 

n  =  Moon's  appar.  motion  in  lat.  in  same  interval  ; 

X'=  Moon's  appar.  lat.  ; 

d  =  Augmented  semi-diameter  of  the  Moon  ; 

6  =  Semi-diam.  of  Sun  ; 

A  =  Appar.  distance  of  centres  of  Sun  and  Moon. 

b  and  c  are  auxiliary  quantities. 

First  find  the  value  of  b  by  the  first  equation,  and  substitute  it  in 
the  second.     Then  derive  the  values  of  c  and  S  from  the  second 


332  ASTRONOMICAL  PROBLEMS. 

and  third  equations,  and  substitute  them  in  the  fourth,  and  it  will 
make  known  the  value  of  £,  which  is  to  be  applied  to  the  approxi- 
mate time  of  the  beginning  of  the  eclipse  according  to  its  sign. 

The  quantities  a,  A;,  n,  &c.,  are  all  to  be  expressed  in  seconds. 
The  apparent  latitude  X'  must  be  affected  with  the  negative  sign 
when  it  is  south.  The  motion  in  latitude,  n,  must  also  have  the 
negative  sign  in  case  the  moon  is  apparently  receding  from  the 
north  pole,  a  and  k  are  always  positive.* 

The  result  may  be  verified,  and  corrected,  by  computing  the  ap- 
parent distance  of  the  centres  at  the  time  found,  and  comparing  it 
with  the  sum  of  the  semi-diameters  minus  5". 

Note.  When  great  precision  is  desired,  the  quantities  k  and  n 
must  be  found  for  some  shorter  interval  than  the  half  duration  of 
the  eclipse.  Let  some  instant  be  fixed  upon,  some  five  or  ten 
minutes  before  or  after  the  approximate  time  of  the  beginning  of 
,,he  eclipse,  according  as  the  contact  takes  place  before  or  after. 
For  this  time  deduce  the  longitude  and  latitude  of  the  moon,  from 
the  longitude  and  latitude  at  the  approximate  time  of  beginning, 
by  means  of  their  hourly  variations  ;  and  thence  calculate  the  ap- 
parent longitude  and  latitude,  and  the  augmented  semi-diameter. 
Find  the  longitude  of  the  sun  for  the  time  in  question,  from  its 
longitude  and  hourly  motion  already  known  for  the  approximate 
time  of  beginning.  Then  proceed  according  to  the  rule  given 
above,  only  using  the  quantities  thus  found  for  the  time  assumed, 
in  place  of  the  corresponding  quantities  answering  to  the  approxi- 
mate time  of  greatest  obscuration.  L  will  always  represent  the 
interval  for  which  k  and  n  are  determined. 

For  the  End. 

Subtract  the  longitude  of  the  sun  at  the  approximate  time  of  the 
end  from  the  apparent  longitude  of  the  moon  at  the  same  time. 
Do  the  same  for  the  approximate  time  of  greatest  obscuration. 
Then  proceed  according  to  the  rule  for  the  beginning,  only  substi- 
tuting everywhere  the  approximate  time  of  the  end  for  the  approx- 
imate time  of  the  beginning,  and  taking  in  place  of  the  formula 
c  =  X'  —  6,  the  following  : 


*  It  will  be  somewhat  more  accurate  to  use  in  place  of  k  and  n,  as  above  de- 

jt  k'  _  k          jf  fci  _  fc 

fined,  the  values  of  the  following  expressions  :  —  —  2$  —  --  —  or  —  —  3$  —  —  —  , 

-s  --  2$  —  —  —  or  —  --  3  A  —  —  —  .    The  first  of  each  of  these  pairs  of  expressions 

D  do  O  00 

is  to  be  used  in  case  the  true  time  of  beginning  is  after  the  approximate  time  ;— 
the  second  in  the  other  case.  A:'  and  «'  are  the  apparent  relative  motions  in  longi- 
tude and  latitude  during  the  last  half  of  L.  In  case  these  expressions  are  used 
the  following  constant  logarithm  is  to  be  employed  instead  of  that  above  given, 
viz.  0.69897. 

In  the  calculation  of  the  end  of  the  eclipse,  k  and  «  will  answer  to  the  last  half 
of  L,  and  k1  and  n'  to  the  first  half. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  383 

For  the  Greatest  Obscuration. 

Take  the  sum  of  the  distances  of  the  moon  from  the  sun  in  ap- 
parent longitude  at  the  approximate  times  of  the  beginning  and  end 
of  the  eclipse,  and  call  it  k.  Take  the  difference  of  the  apparent 
latitudes  of  the  moon  at  the  same  times,  if  the  two  are  of  the  same 
name  ;  but  if  they  are  of  different  names,  take  their  sum.  Denote 
the  difference  or  sum  by  n.  Let  a'  =  the  distance  of  the  moon 
from  the  sun  in  apparent  longitude  at  the  true  time  of  greatest  ob- 
scuration ;  X'  =  the  apparent  latitude  of  the  moon  at  the  approxi- 
mate time  of  greatest  obscuration. 

k  :  n  :  :  X'  :  a'. 

Find  the  value  of  a'  by  this  proportion,  affecting  X',  n,  k,  always 
with  the  positive  sign. 

Ascertain  whether  the  greatest  obscuration  has  place  before  or 
after  the  apparent  conjunction,  by  observing  whether  the  apparent 
latitude  of  the  moon  is  increasing  or  decreasing  about  this  time , 
the  rule  being,  that  when  it  is  increasing,  the  greatest  obscuration 
will  occur  before  apparent  conjunction ;  when  it  is  decreasing, 
after.  If  the  approximate  and  true  times  of  greatest  obscuration 
are  both  before  or  both  after  apparent  conjunction,  from  the  value 
found  for  a'  subtract  the  distance  of  the  moon  from  the  sun  in  ap- 
parent longitude  at  the  approximate  time ;  but  if  one  of  the  times 
is  before  and  the  other  after  apparent  conjunction,  take  the  sum  of 
the  same  quantities.  Denote  the  difference  or  sum  by  in.  Also, 
let  D  =  duration  of  eclipse,  and  t  =  correction  to  be  applied  to  the 
approximate  time  of  greatest  obscuration.  Then  to  find  t,  we  have 
the  proportion 

k  :  m  :  :  D  :  t. 

If  the  apparent  latitude  of  the  moon  is  decreasing,  t  is  to  be 
applied  according  to  the  sign  of  m  ;  but  if  the  apparent  latitude  is 
increasing,  it  is  to  be  applied  according  to  the  opposite  sign. 

A  still  more  exact  result  may  be  had  by  repeating  the  foregoing 
calculations,  making  use  now  of  the  apparent  latitude  at  the  time 
just  found.  When  the  greatest  accuracy  is  required,  the  values  of 
k  and  n  may  be  found  more  exactly  after  the  same  manner  as  foi 
the  beginning  or  end. 

For  the  Quantity  of  the  Eclipse. 

Find  by  interpolation  the  apparent  latitude  of  the  moon  at  the 
true  time  of  greatest  obscuration.  With  this,  and  the  distance  in 
longitude  a'  obtained  by  the  proportion  above  given,  compute  by 
the  formulae  on  page  375,  the  apparent  distance  of  the  centres  of 
the  sun  and  moon  at  the  time  of  greatest  obscuration.  Subtract 
this  distance  from  the  sum  of  the  apparent  semi-diameters  of  the 


384  ASTRONOMICAL  PROBLEMS. 

two  bodies,  diminished  by  5",  and  denote  "the  remainder  by  R 
Then, 

Sun's  semi-diam.  (diminished  by  3") :  R  :  :  6  digits  :  number  ot 
digits  eclipsed. 

When  the  apparent  distance  of  the  centres  of  the  sun  and  moon 
at  the  time  of  greatest  obscuration  is  less  than  the  difference  be- 
tween the  sun's  semi-diameter  and  the  augmented  semi-diameter 
of  the  moon,  the  eclipse  is  either  annular  or  total ;  annular,  when 
the  sun's  semi-diameter  is  the  greater  of  the  two ;  total,  when  it 
is  the  less. 

For  the  Beginning  and  End  of  the  Annular  or  Total  Eclipse. 

The  times  of  the  beginning  and  end  of  the  annular  or  total 
eclipse  may  be  found  as  follows  :  the  greatest  obscuration  will  take 
place  very  nearly  at  the  middle  of  the  eclipse  in  question,  and  will 
not  differ,  at  most,  more  than  five  or  eight  minutes  (according  as 
the  eclipse  is  total  or  annular)  from  the  beginning  and  end  :  to 
obtain  the  half  duration  of  the  eclipse,  and  thence  the  times  of  the 
beginning  and  end,  we  have  the  formulae 

log.  tang  d  —  log.  X' +ar.  co.  log.  a,  log.  k'=log.  k  +  ar.  co.  log.  sin  0 ; 
S  =  8-d-  1",  orS=d  — £  +  1"; 
_ log.(S+A)  +  log.(S— A)  ^ 

JOg.  C  -  2  9 

log.  t  =  ar.  co.  log.  V  +  log.  c  +  log.  D  +  1 .77815  —  10 ; 

Time  of  Begin.  =  M  —  t,  Time  of  End  =  M-f  t : 
in  which 

M  =  Time  of  greatest  obscuration  ; 

X'  =  Moon's  apparent  latitude  at  that  time  ; 

a  =  Distance  of  moon  from  sun  in  appar.  long. ; 

k  =  Variation  of  this  distance  during  the  whole  eclipse,  or  rela 

tive  mot.  in  appar.  long,  during  this  interval ; 
k'  =  Moon's  appar.  mot.  on  relative  orbit  for  same  interval ; 
6    =  Inclination  of  relative  orbit ; 
6    —  Semi-diameter  of  sun  ; 
d  =  Augm.  semi-diam.  of  moon  ; 
A  =  Appar.  distance  of  centres ; 

D  =  Duration  of  eclipse,  (partial  and  annular  or  total ;) 
t    =  Half  duration  of  annular  or  total  eclipse. 

The  fast  value  of  S  is  used  when  the  eclipse  is  annular,  the 
second  when  it  is  total.  The  quantities  may  all  be  regarded  as 
positive.  The  results  may  be  verified  and  corrected  by  finding 
directly  the  apparent  distance  of  the  centres  for  the  times  obtained, 
and  comparing  it  with  the  value  of  S. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  385 

For  the  Point  of  the  Surfs  Limb  at  which  the  Eclipse  commences. 

Find  the  angle  of  position  of  the  sun,  and  the  angle  between  its 
vertical  circle  and  circle  of  declination,  at  the  beginning'  oi  the 
eclipse,  as  explained  at  page  380.  Let  the  former  be  denoted  by 
p,  and  the  latter  by  v.  Give  to  each  the  negative  sign,  if  laid  off 
towards  the  right  ;  the  positive  sign  if  laid  off  towards  the  left. 
Let  a  =•  distance  of  the  moon  from  the  sun  in  apparent  longitude 
at  the  beginning  of  the  eclipse  ;  X'  =  the  moon's  apparent  latitude 
at  the  same  time  ;  and  &  =  angular  distance  of  the  point  of  contact 
from  the  ecliptic.  Compute  the  angle  &  by  the  formula 

log.  tang  6  •==•  log.  X'  +  ar.  co.  log.  a  ; 

taking  it  always  less  than  90°,  and  positive  or  negative  according 
to  the  sign  of  its  tangent.  X'  is  negative  when  south  ;  a  is  always 
positive. 

Let  A  =  distance  on  the  limb  of  the  point  of  contact  from  the 
vertex.  The  above  operations  being  performed,  the  value  of  A 
results  from  the  equation 


p,  v,  and  d  being  taken  with  their  signs. 

If  the  result  is  affected  with  the  positive  sign,  the  point  first 
touched  will  lie  to  the  right  of  the  vertex.  If  with  the  negative 
sign,  it  will  lie  to  the  left  of  the  vertex. 

Note.  The  circumstances  of  an  occultation  of  a  fixed  star  by 
the  moon  may  be  calculated  in  nearly  the  same  manner  as  those 
of  a  solar  eclipse.  The  star  in  the  occultation  holds  the  place  of 
the  sun  in  the  eclipse.  The  immersion  and  emersion  of  the  stai 
correspond  to  the  beginning  and  end  of  the  eclipse.  The  elements 
which  ascertain  the  relative  apparent  place  and  motion  of  the  moon 
and  star,  take  the  place  of  those  which  ascertain  the  relative  appa- 
rent place  and  motion  of  the  moon  and  sun.  Thus  the  star's  lon- 
gitude, corrected  for  aberration  and  nutation,  (see  Problem  XXIII,) 
must  be  used  instead  of  the  sun's  longitudes  ;  the  apparent  dis- 
tances of  the  moon  from  the  star  in  latitude,  instead  of  the  moon's 
apparent  latitudes  ;  and  the  moon's  augmented  semi-diameter,  in- 
stead of  the  sum  of  the  semi-diameters  of  the  sun  and  moon.  The 
difference  of  the  longitudes,  and  the  relative  motion  in  longitude. 
must  also  now  be  reduced  to  a  parallel  to  the  ecliptic  passing 
through  the  star,  (see  Appendix,  page  431.)  If  X  =  apparent  lati- 
tude of  star,  a  =  diff.  of  appar.  longitudes  of  moon  and  star,  and 
k  —  relative  motion  in  longitude,  we  must  substitute  in  the  formu- 
la for  the  eclipse,  for  X',X'  —  X  ;  for  a,  a  cos  X  ;  and  for  k,  k  cos  X. 
n  will  stand  for  the  relative  motion  in  latitude,  or  for  the  variation 
of  x'  —  X. 

Example.  Required  to  calculate  an  eclipse  of  the  sun,  for  the 

25 


386 


ASTRONOMICAL  PROBLEMS. 


latitude  and  meridian  of  New  York,  that  will  occur  on  the  18th  of 
September,  1838. 

For  the  Approximate  Times  of  the  Phases. 

Approximate  time  of  New  Moon. 

Sept.  18d-  8h-  49m- 


Sun's  longitude,  . 

. 

175°  27' 

31".4 

Do.  hourly  motion,      .         • 

2 

26  .7 

Do.  semi-diameter,      .        • 

15 

57  .0 

Moon's  longitude, 

. 

. 

175  29 

19 

Do.  latitude, 

. 

. 

f.            47 

47 

Do.  equatorial  parallax,         . 

53 

53 

Do.  semi-diarneter, 

14 

41 

Do.  hor.  mot.  in  long. 

29 

29 

Do.  hor.  mot.  in  lat.    .         . 

2 

41 

Do.  appar.  long.  (Prob.  XVII), 

175   10 

26 

Do.  appar.  lat.  (X7), 

• 

2 

25  N. 

Do.  augm.  semi-diameter,    . 

14 

47 

DifF.  of  appar.  long,  (a),       . 

>           !••:. 

17 

5 

Appar.  dist.  of  cen 

.  (A), 

. 

, 

17 

15 

Sum  of  semi-diameters,        . 

r            30 

44 

7h.  49™. 

Sun's  longitude,  . 

175°  25' 

4" 

Moon's  appar.  long.     . 

.     174  47 

3 

Do.  appar.  lat.  (X7) 

. 

8 

12  N. 

Do.  augm.  semi-diameter, 

14 

49 

DifF.  of  appar.  long,  (a), 

38 

1 

Appar.  dist.  of  cen.  (A), 

.         .         .             38 

53 

Sum  of  semi-diameters, 

30 

46 

gh.  49n.. 

Sun's  longitude,  . 

j 

.     175°  29' 

58" 

Moon's  appar.  long.     . 

.     175  36 

15 

Do.  appar.  lat.  (X'), 

2 

18  S. 

Do.  augm.  semi-diameter, 

.        .             14 

44 

DifF.  of  appar.  long,  (a), 

.        vY     .               6 

17 

Appar.  dist.  of  cen 

.  (A), 

•     .*'••"•            6 

42 

Sum  of  semi-diameters, 

30 

41 

a        diff.  or  k. 

X' 

diff.  or  n 

A            diff. 

sum  semi-d. 

7M9-    2281" 
8    49       1025 

492"  N 
145   N 

347" 

2333"          Rl/ 
1035 

1846" 
1844 

9    49         377      Jrr' 

138    S 

283 

402      1CC£. 

1841 

10  49       1925      l 

357    S 

219 

1958      1556 

1839 

TO  CALCULATE  A  SOLAR  ECLIPSE. 

For  the  Approximate  Time  of  Beginning. 
k  =  1298",  d  =  2333"  —  1846"  =487"  ; 

1298"  :  487"  :  :  GO*-  :  t  =  22m-.5 
7h.49m. 

22 


387 


1st  Approxi.  &•  1 

7h-49m-      .      a  =  2281" 
Corrections  for  22™-  447 

gb.  nm.      m      a  =  i834         .       ? 

a  =  1 834"  ar.  co.  log.  6.73660  . 
X'  =  359   .   log.  2.55509 

6  =11°  4'  30".  tan.  9.29169  , 

Appar.  dist.  of  cen.       A  =  1 869" 
Sum  of  semi-diam.        .     1846 


X'=492"N. 

133  (See  Note,  p.  324) 

v  =  359   N. 

.        .        log.  3.26340 

ar.  co.  cos.  0.00817 


.     log.  3.27157 


487"  :    23"  :  :  22™-  :  t 

gh. 


2d  Approxi.  &*-  12"- 
For  the  Approximate  Time  of  the  End. 

h  =  1556",  d  =  1958"  -  1839"  =  119". 

1556"  :  119":  :  60m-  :  t  =  4nu.6. 
10h. 

-5 


1st  Approxi.  1011-  44^ 

49™-      .      a  =  1925"       .         . 
Corrections  for  5m-    132        .        . 


10h.  44m. 


a  =  1793 


a 

X' 


1793"  .   ar.  co.  log.  6.74642 
340     .    .  log.  2.53148 


V  =  357"  S. 
17 

X'  =  340  S. 
log.  3.25358 


.  tan.  9.27790  .  ar.  co.  cos.  0.00767 


Appar.  dist.  of  cen.  A  =  1825" 
1839 


3.26125 
133"  :       14'-'  :  :  5m- :  t  =  O^.S 


388 


ASTRONOMICAL   PROBLEMS. 


10h.  440,. 

0  .5 

SdApproxi.  !&•  44m\5 
For  the  Approximate  Time  of  Greatest  Obscuration 


Approx.  time  of  begin.     .      &*• 
Approx.  time  of  end,         .     10    44 

2  )  18    56 

IstApproxi.     .     9     28 

For  the  True  Times  of  the  Phases. 


Approx.  time  of 
Beginning. 

Qh.  jgm. 


Approx.  time  of 
Greatest  Obscur. 

gh.  28m. 


Approx.  time  of 
End. 

]0h.  44™. 


Sun's  longitude,175°  26'    1".0      175°  29'    6".8    175°32' 12".6 
Do.  semi-diam.,         15  57  .0  15  57  .0  15  57  .0 

Moon's  app.  Ion.  174  55  36  .7      175  27     7  .7    176     2  17  .2 
Do.  app.  lat.  5  45  .3N.  0  43  .58.  5  32  .48 

Do.augm.semid.       14  48  .0  14  45  .1  14  41  .7 


Qh.  jgm. 

9  28 
10  44 

a 

k 

A<    ,   « 

A 

8 

1824".3 
119  .1 
1804  .6 

1705".2 
1923  .7 

345;/.3  N 
43  .58 
332  .48 

388".8 
288  .9 

1856".7 
1835  .0 

1840".0 
1833  .7 

For  the  True  Time  of  Beginning. 


1824".3 
1705  .2 

388  .8 


I  =  -  8001  .1 
X'     345  .3 


.  log.  3.26109 
.  log.  3.23178 
ar.  co.  log.  7.41028— 

.  log.  3.90315— 


S  +  A  .  3696  .7   . 

8- A  .  -16  .7   . 

n  .  . 

L  .  76m. 


Corr.  of  approx.  time,          +  43'- .4 


ar.  co.  log.  6.07850 
.  log.  3.56781 
.  log.  1.22272— 

ar.  co.  log.  7.41028— 
.  log.  1.88081 

Const,  log.  1.47712 

.     log.  1.63724 -f 


TO  CALCULATE  A  SOLAR  ECLIPSE.  389 

Corr.  of  approx.  time,  -f  43s- .4 

Approx.  time,         .      &*•  12^  0  .0 

True  time  of  begin.     8    12   43  .4,  in  Greenwich  time 
Diff  of  merid.  4    56      4 


True  time  of  begin.  3    16    39  .4,  in  New  York  time. 

For  the  True  Time  of  End. 

a      .     .     1804".6  ....  log.  3.25638 

k      .     .     1923  .7  ....  log.  3.28414 

n      .     .      288  .9  -.        .         ar.  co.  log.  7.53925— 

b  =      -  12016  .3  ....  log.  4.07977— 
V             —332  .4 


X'  +6=c=  -12348  .7        .        .        ar.  co.  log.  5.90838— 
S+A     .    .      3668.7        .        .         .        .log.  3.56451 
S-A     .    .        —1.3        .        .        .        .  log.  0.11394— 
n     .     .        .         .        .        •         ar.  co.  log.  7.53925— 
L    .    .        .    76m.      ....  log.  1.88081 

Const,  log.  1.47712 

Corr.  of  approx.  time,  —  3"-  0        .  log.  0.48401  — 

Approx.  time,         .     10*-  44nu  0  .0 

True  time  of  end,  .     10    43   57  .0,  in  Greenwich  time. 
Diff.  of  merid.  4     56     4 


True  time  of  end,  .       5     47   53,       in  New  York  time. 

For  the  True  Time  of  Greatest  Obscuration. 

True  time  of  beginning,  &••  12m-43g-.4 

Do.  of  end,     .        .        .        .     10    43    57    .0 


2)  18     56    40    .4 

2d  Approx.     9     28    80   A 

&-  49BB-    .        .    X'  =  138"     S. 
9     28       .        .    V  =  43  .5  S. 

Diff.   21  Diff.  94  .5 

21"1-  :  201-  :  :  94".5  :  1".5 
43  .6 

9k-  28™-  20*  .  V  =  45.  0 


390  ASTRONOMICAL  PROBLEMS. 

1705".2  388".8 

1923  .7  288  .9 


k  =  3628  .9       :  n  =  677  .7  : :  Xf  =  45".0  :  a'  =  8".4 

Time  of  beginn.  8h-  12m-  43s-  .4,  at  9h-  28"1-  a  =  119".l 
Time  of  end,     10    43     57.0  o'  =     8.4 


D=   2    31     13  .6  m  =  -  110  .7 


3628".9  :  110".7  :  :  2h-  31m-  13s- .6  :  4m-  36s-  .8 

9h-28       0  .0 


True  time  (nearly)  9  32    36  .8 


43  .5 
At  9h- 32m- 37%  X'=64  .3 

3628".9  :  677".7 : :  64;/.4  : 12".0 ;  at  9h>  32ra-  37s-,  a  =  8".4 

a'=12  .0 

m  =  3  .6 

8628".9 :  3".6  : :  2h-  31m-  13s-.6 :      9s* .0 

9h-  32m-  36  .8 


9   32     27.8 

True  time  of  greatest  obscur.    .    9h-  32m-  27s-. 8,  in  Greenw.  time 
Diff.  of  merid.  .  ,    4    56      4 


True  time  of  greatest  obscur.    .    4    36    23  .8,  in  N.  Y.  time. 
For  the  Quantity  of  the  Eclipse. 
9h.  32m.  3^s.     ^    x'  =  64/;.3 

At  nearest  approach  of  centres,    .    X'  =  63  .7 
"  "        "       .        *     a  =  12  .0 

a     .     12".0     .      ar.  co.  log.  8.92082,    .        .    log.  1.07918 
X'    .     63  .7    .        .  .      .      1.80414 

6     .      v;;      .       V      tan.  0.72496,    .  ar.  co.  cos.  0.73253 

Shortest  distance  of  centres,    64".8    .        ;    log.  1.81171 
Sum  of  semi-diameters,       1837  .0 

1772  .2 
15'  54"  :  1772".2  :  :  6  :  11.14  digits  eclipsed. 


TO  CALCULATE  A  SOLAR  ECLIPSE.  391 

For  the  Situation  of  the  Point  at  which  the  Obscuration  com- 
mences. 
8h-  12™-     .     .     a     =1824",      .         .         X'  =  345''.3N 

76m.  .  438.   .   .  17Q5//  .  jg^  76m.   .  438.  .  . 


At  the  beginn.      .        a  =1808,      .         . 
a     .     1808       .     or.  co.  log.  6.74280 
X'    .       341.6       .      .      log.  2.53352 

6  =  10°  41'  57"     .     .       tan.  9.27632 

Obliq  eclip.  (Prob.X),  23°  27' 47"  .sin.  9.60005  .  tan.  9.63753 

Sun's  longitude,          175   26     3    .  sin.  8.90093  .  cos.  9.99862- 

sin.  8.50098,     tan.  9.63615  — 

Sun's  declination,  1°  49'  0/; ;  Angle  of  pos.  23°  23'  50". 
Meantime  of  begin.  3h-  16m-  39s-,  Lat.  40°  42'  40",  Dec.  1°49'  0" 
Equa.  of  time,  5     58  90  90 

Appar.time,     .         3    22     37,  PZ  =49  17   20,  PS  =88  11 
60 


4 ) 202         37 

Hour  angle  P  =  50°  39'  15"         .         cos.  9.80210 
Co.  lat.     PZ  =  49  17  20  tan.  0.06526 


m  =  36°  23'    0"       .         .         tan.  9.86736 
Co.  dec.  PS  =88   11     0 


m'=51   48     0         .        ar.  co.  sin.  0.1 0466 
m  =  36   23     0         .         .  sin.  9.77320 

P=  50  39  15         .         .          tan.  0.08627 

S=  42  38  10         .         .          tan.  9.96413 
Angle  of  position,  .         .         —  23°  23'  50" 

Angle  from  eclip.  (6),         .         .         —  10  41  50 
Angle  of  dec.  circle  from  vertex  (S),       42  38  10 

90 


Angular  dist.  of  point  first  touched  from  vertex,  98  32,  to  the  right 
For  the  Beginning  and  End  of  the  Annular  Eclipse, 

Approx.  time,  9h-  32m-  27s-. 8  =  true  time  of  greatest  obscur. 

At  this  time,  a  =  12". 2,  X'  =  63".7. 

a  =  12". 2         .          ar.  co.  log.  8.91364     .         .     log.  1.08636 
X'=63  .7         .         .  log.  1.80414 

a  =  79°  9'  30"  .  tan.  0.71778    .    ar.  co.  cos.  0.72564 

A  =  64".9      .         .     log.  1.81200 


S92  ASTRONOMICAL  PROBLEMS. 

S  +  A  =  135".8  .  log.  2.13290, 6  =79°  9'  30"  .  ar.  co.  sin.  0.00783 
S  -  A  =  6  .2  .  log.  0.79239,  &=3628".9   .      log.  3.55977 

2  )  2.92529,  V  .    .    ar.  co.  log.  6.43240 

1.46264          ....         1.46264 
D=152m-          .         .         .         .log.  2.18184 

Const,  log.  1.77815 

i==0h.  lm.  1P.  6          ^  ,        iog.  h 85503 
Time  of  greatest  obscur.   .  4  36    23  .8 

Formation  of  ring,     .         .    4  35    12  .2,  New  York  time. 
Rupture  of     do.       .        .4  37    35  .4 


PROBLEM  XXXI. 

To  find  the  Moorfs  Longitude,  Latitude,  Hourly  Motions,  Equa- 
torial Parallax,  and  Semi-diameter,  for  a  given  time,  from  the 
Nautical  Almanac* 

Reduce  the  given  time  to  mean  time  at  Greenwich ;  then, 
For  the  Longitude. 

Take  from  the  Nautical  Almanac  the  calculated  longitudes  an- 
swering to  the  noon  and  midnight,  or  midnight  and  noon,  next  pre- 
ceding and  next  following  the  given  time.  Commencing  with  the 
longitude  answering  to  the  first  noon  or  midnight,  subtract  each 
longitude  from  the  next  following  one  :  the  three  remainders  will 
be  the  first  differences.  Also  subtract  each  first  difference  from 
the  following  for  the  second  differences,  which  will  have  the  plus 
or  minus  sign,  according  as  the  first  differences  increase  or  de- 
crease. 

Find  the  quantity  to  be  added  to  the  second  longitude  by  rea- 
son of  the  first  differences,  by  the  proportion,  1 2b< :  excess  of  given 
time  above  time  of  second  longitude  :  :  second  first  difference : 
fourth  term. 

With  the  given  time  from  noon  or  midnight  at  the  side,  take  from 
Table  XCIII  the  quantities  corresponding  to  the  minutes,  tens  of 
seconds,  arid  seconds,  of  the  mean  or  half  sum  of  the  two  second 
differences,  at  the  top  :  the  sum  of  these  will  be  the  correction  for 
second  differences,  which  must  have  the  contrary  sign  to  the  mean. 

The  sum  of  the  second  longitude,  the  fourth  term,  and  the  cor 
rection  for  second  differences,  will  be  the  longitude  required. 


TO  FIND  MOON'S  LONG.,  ETC.,  FROM  NAUTICAL  ALMANAC.     393 


For  the  Latitude. 

Prefix  to  north  latitudes  the  positive  sign,  but  to  south  latitudes 
the  negative  sign,  and  proceed  according  to  the  rules  for  the  lon- 
gitude, only  that  attention  must  now  be  paid  to  the  signs  of  the  first 
differences,  which  may  either  be  plus  or  minus. 

The  sign  of  the  resulting  latitude  will  ascertain  whether  it  is 
north  or  south. 

For  the  Hourly  Motion  in  Longitude. 

Solve  the  proportion,  1 2h-  :  given  time  from  noon  or  midnight 
.  :  half  sum  of  second  differences  :  a  fourth  term ;  which  must  have 
the  same  sign  as  the  half  sum  of  the  second  differences. 

Take  the  sum  of  the  second  first  difference,  half  the  mean  of 
the  second  differences,  with  its  sign  changed,  and  this  fourth  term, 
and  divide  it  by  12  :  the  quotient  will  be  the  required  hourly  mo- 
tion in  longitude. 

For  the  Hourly  Motion  in  Latitude. 

With  the  given  time  from  noon  or  midnight,  the  second  first 
difference  of  latitude,  and  the  mean  of  the  second  differences,  find 
the  hourly  motion  in  latitude  in  the  same  manner  as  directed  for 
finding  the  hourly  motion  in  longitude.  When  the  hourly  motion 
is  positive,  the  moon  is  tending  north ;  and  when  it  is  negative, 
she  is  tending  south. 

For  the  Semi-diameter  and  Equatorial  Parallax. 

The  moon's  semi-diameter  and  equatorial  parallax  may  be  taken 
from  the  Nautical  Almanac,  with  sufficient  accuracy,  by  simple 
proportion,  the  correction  for  second  differences  being  too  small  to 
be  taken  into  account,  unless  great  precision  is  required. 

Corrections  for  Third  and  Fourth  Differences. 

When  the  moon's  longitude  and  latitude  are  required  with  great 
precision,  corrections  must  also  be  applied  for  the  third  and  fourth 
differences.  To  determine  these,  take  from  the  Almanac  the  three 
longitudes  or  latitudes  immediately  preceding  the  given  time,  and 
the  three  immediately  following  it,  and  find  the  first,  second,  third, 
and  fourth  differences,  subtracting  always  each  number  from  the 
following  one,  and  paying  attention  to  the  signs.  With  the  given 
time  from  noon  or  midnight  at  the  side,  and  the  middle  third 
difference  at  the  top,  take  from  Table  XCIV  the  correction  for 
third  differences,  which  must  have  the  same  sign  as  the  middle 
third  difference  when  the  given  time  from  noon  or  midnight  is  less 
than  6  hours  ;  the  contrary  sign,  when  the  given  time  is  more  than 
6  hours. 


394  ASTRONOMICAL  PROBLEMS. 

With  the  given  time,  and  half  sum  of  fourth  differences,  take 
from  Table  XCV  the  correction  for  fourth  differences,  giving  it 
always  the  same  sign  as  the  half  sum. 

The  sum  of  the  third  longitude  or  latitude,  the  proportional  part 
of  the  middle  first  difference  answering  to  the  given  time  from 
noon  or  midnight,  and  the  corrections  for  second,  third,  and  fourth 
differences,  having  regard  to  the  signs  of  all  the  quantities,  will  be 
the  longitude  or  latitude  required. 


APPENDIX. 


TRIGONOMETRICAL    FORMULAE." 

I.  RELATIVE  TO  A  SINGLE  ARC  OR  ANGLE  a. 

1.  sin2  a  +  cos2  a  =  I 

2.  sin  a  =  tan  a  cos  a 

tan  a 

3.  sin  a  = 

4. 

5.  tan  a  = 

6.  cot  a  = 


\-  tan2  a 

1 


+  tan2  a 
sin  a 

cos  a 

1      _  cos  a 
tan  a      sin  a 

7.  sin  a  =  2  sin  £  a  cos  £  a 

8.  cos  a  =  1  —  2  sin2  £  a 

9.  cos  a  =  2  cos2  ^  a  —  1 


-IV. 

UUl    2 

1 

T  cos  a 

11. 

cot  | 

a^— 

sin  a 

1 

—  cos  a 

19. 

tan2  I 

1 

—  cos  a 

1      cos  a 

13.  sin  2  a  =  2  sin  a  cos  a 

14.  cos  2  a  =  2  cos2  a  —  1  =  1  —  2  sin1  a 

II.  RELATIVE  TO  Two  ARCS   a  AND  6,  OF  WHICH  a  is  SUPPOSED 

TO  BE  THE  GREATER. 

15.  sin  (a  +  b)  =  sin  a  cos  b  +  sin  6  cos  a 

16.  sin  (a  —  b)  =  sin  a  cos  6  —  sin  b  cos  a 

17.  cos  (a  +  b)  =  cos  a  cos  6  —  sin  a  sin  6 

•  The  radius  is  supposed  to  be  equal  to  unity  in  all  of  the  formulae. 


396  APPENDIX. 

18.  cos  (a  —  b)  ~  cos  a  cos  b  +  sin  a  sin  b 

,  7  x       tan  a  +  tan  b 

19.  tan  (a  +  b)  = — 7 

1  —  tan  a  tan  b 

T.      tan  a  —  tan  6 

20.  tan  (a  —  6)  =  — r- — 7 

l+tanatano 

21.  sin  a  +  sin  b  —  2  sin  J  (a  +  6)  cos  \(a  —  5) 

22.  sin  a  —  sin  6  =  2  sin  £  (a  —  b)  cos  J  (a  +  b) 

23.  cos  a  -4-COS&  =  2  cos  J  (a  +  6)  cos  £  (a  —  b) 

24.  cos  b  —cos a  =  2  sin  %  (a  +  b)  sin  £  (a  —  b) 

sin  (a 

25.  tan  a  +  tan  b  =  — *- 


— — -- r 
cos  a  cos  6 

,       sin  (a  —  b) 

26.    tan  a  —  tan  &  = \ 1 

cos  a  cos  6 


»/. 

28. 
29. 
30. 
31. 

32. 

33. 
34. 
35. 

36. 
37 
38. 
39. 
40. 

coi  a  -t-  coi  o  :  .  —  7 
sin  a  sm  b 

,                    sin  (a  —  b) 

6) 

sm  a  sm  6 
sin  a  +  sin  b      tan  |  (a  +  b) 

sin  a  —  sin  b      tan  }  (a  —  b) 
cos  6  +  cos  a      cot  |  (a  +  b) 

cos  6  —  cos  a      tan  |  (a  —  b) 
tan  a  +  tan  b      cot  6  +  cot  a      sin  (a  + 

tan  a  —  tan  6      cot  6  —  cot  a      sin  (a  — 
cot  b  —  tan  a      cot  a  —  tan  b      cos  (a  + 

6) 
b) 

cot  6  +  tan  a      cot  «  +  tan  b      cos  (a  — 
sin2  a  —  sin2  b  —  sin  (a  +  6)  sin  (a  —  6) 
cos2  a  —  sin2  b  =  cos  (a  +  6)  cos  (a  —  6) 
1  ±  sin  a  =  2  sin2  (45°  ±  £  a) 

b) 

1  ^F  sin  a 

cos  a 
1  —  sin  a      sin2  (45°  —  i  a) 

1  —  cos  a              sin2  £  a 
1  -f  sin  6      sin2  (45°  +  J  fc) 

1  +  cosa            cosa£a 

[=4|  =  tan(45°-&) 

TRIGONOMETRICAL    FORMULA. 


397 


42.  sin  a  cos  b  =  1  sin  (a  +  b)  -f  £  sin  (a  —  6) 

43.  cos  a  sin  b  =  |  sin  (a  +  b)  —  |  sin  (a  —  6) 

44.  sin  a  sin  6  =  j  cos  (a  —  6)  —  J  cos  (a  +  b) 

45.  cos  a  cos  6  =  j  cos  (a  +  b)  + 1  cos  (a  —  6) 

III.  TRIGONOMETRICAL  SERIES. 

a3  a5 

Sln  *  =«  __  +  ^-^  -&c. 


46. 


cos  a  =  1  — 


tan  a 


&c. 


Let  <z  =  length  of  an  arc  of  a  circle  of  which  the  radius  is  1,  and 
(a")  =  number  of  seconds  in  this  arc,  then  to  replace  an  arc  ex- 
pressed by  its  length,  by  the  number  of  seconds  contained  in  it,  we 
have  the  formula 

47.  a  =  (a")  sin  I"  ;  log.  sin  I"  =^6.685574867. 

IV.  DIFFERENCES  OF  TRIGONOMETRICAL  LINES. 

48.  A  sin  x  =  +  2  sin  |  A  x.  cos  (x  +  ±  A  x) 

49.  A  cos  x  =  —  2  sin  %  A  x.  sin  (x  +  i  A  #  ) 

sin  A  a? 

50.  A  tan  a:  =  -f 


51.    A  cot  x  —  —  -. 


.      , 
cos  a?,  cos  (#  4- 

sin  A  x 


. 
sin  x.  sin  (x  +  A  x) 

V.  RESOLUTION  OF  RIGHT-ANGLED  SPHERICAL  TRIANGLES  * 
Table  of  Solutions. 

Given.  Required.  Solution. 

Hypothen.  f  side  op.  giv.  ang.  52    sin  x  =  sin  h  .  sin  a 

and      <  side  adj.  giv.  ang.   53    tan  x  —  tan  h  .  cos  a 

an  angle    L  tne  other  angle       54     cot  x  =  cos  h  .  tan  a 

{,       .1        .,  cos  A 

the  other  side         55     cos  x  =  - 
cos  s 
ang.  adj.  giv.  side  56     cos  x  =  tan  s  .col  h 
.,  .  sin  s 

ang.  op.  giv.  side  57    sm  x  =  -  —  ? 


*  Baily's  Astronomical  Tables  and  Formate. 


398  APPENDIX. 


...        j     the  hypothen. 
A  side  and  | 

the  angle  J  the  other  side 

opposite 
the  other  angle 

A  side  and  f  the  hypothen. 
the  angle  <  the  other  side 
adjacent    ^  the  other  angle 
f  the  hypothen. 
The  two  < 
sides        [  an  angle 

f  the  hypothen. 
The  two   1 

ansles    U-iit 

58 
59 
60 

61 
62 
63 
64 

65 
66 

RT 

sm  s          "]  i 

sm  x  =  —  
sm  a 

sin  a;  =  tan  s  .  cot  a  y  | 
cosa 

cos  *          J  | 
cot  #  =  cos  a  .  cot  s 
tana;  =  tan  a  .  sin  s 
cos  a;  =  sin  a  .  cos  5 
cos  x  =  rectang.  cos.  ( 
giv, 
cot  a?  =  sin  adj.  side   : 

°1 

cos  x  =  rectang.  cot.  < 
given  ; 
cos.  opp.  ang. 

of  the 
sidea 
x  cot. 
side 
of  the 


sm.  adj.  ang. 

In  these  formulae,  x  denotes  the  quantity  sought. 
a  =  the  given  angle 
*  =  the  given  side 
h  =  the  hypothenuse. 


The  formulae  for  the  resolution  of  right-angled  spherical  trian- 
gles are  all  embraced  in  two  rules  discovered  by  Lord  Napier,  and 
called  Napier's  Rules  for  the  Circular  Parts.  The  circular  parts, 
so  called,  are  the  two  legs  of  the  triangle,  or  sides  which  form  the 
right  angle,  the  complement  of  the  hypothenuse,  and  the  comple- 
ments of  the  acute  angles.  The  right  angle  is  omitted.  In  re* 
solving  a  right-angled  spherical  triangle,  there  are  always  three  of 
the  circular  parts  under  consideration,  namely,  the  two  given  parts 
and  the  required  part.  When  the  three  parts  in  question  are  con- 
tiguous to  each  other,  the  middle  one  is  called  the  middle  part,  and 
the  others  the  adjacent  parts.  When  two  of  them  are  contiguous, 
and  the  third  is  separated  from  these  by  a  part  on  each  side,  the 
part  thus  separated  is  called  the  middle  part,  and  the  other  two  the 
opposite  parts.  The  rules  for  the  use  of  the  circular  parts  are  (the 
radius  being  taken  =  1 ), 

1.  Sine  of  the  middle  part  =  the  rectangle  of  the  tangents  of  the 
adjacent  parts. 

2.  Sine  of  the  middle  part  =  the  rectangle  of  the  cosines  of  the 
opposite  parts. 

PARTICULAR  CASES  OF  RIGHT-ANGLED  SPHERICAL  TRIANGLES. 

Equations  52  to  67,  or  Napier's  rules,  are  sufficient  to  resolve 
all  the  cases  of  right-angled  spherical  triangles  ;  but  they  lack  pre- 
cision if  the  unknown  quantity  is  very  small  and  determined  by 


RESOLUTION  OF  SPHERICAL  TRIANGLES.  399 

means  of  its  cosine  or  cotangent;  or,  if  the  unknown  quantity  is 
near  90°,  and  given  by  a  sine  or  a  tangent  :  in  these  cases  the  fol- 
lowing formulae  may  be  used  : 


. 

cos(B  —  C) 

69.  tan'  IB  =^^| 

sm  (a  +  c) 

70.  tan2  \c  =  tan  \  (a  +  b)  tan  £  (a  —  b) 

71.  tan  (45°  —  16)  =  ^  tan  (45°—  a?),  tan  x  =  sin  a  sin  B 

72.  tan"  \b  =  tan  (5^5  +  45°)  tan  (*L±_<?  -45°). 

a  is  the  hypothenuse,  B,  C,  the  acute  angles,  and  6,  c,  the  sides 
opposite  the  acute  angles. 

VI.  RESOLUTION  OF  OBLIQUE-ANGLED  SPHERICAL  TRIANGLES. 

General  Formula. 

Let  A,  B,  C,  denote  the  three  angles  of  a  spherical  triangle,  and 
a,  b,  c,  the  sides  which  are  opposite  to  them  respectively. 
sin  A  _  sin  B  _  sin  C 
sin  a       sin  6      sin  c 

or,  the  sines  of  the  angles  are  proportional  to  the  sines  of  the  op- 

posite sides. 

74.  cos  c  —  cos  a  cos  b  +  sin  a  sin  b  cos  C 

75.  cos  c  =  cos  (a  —  6)  —  2  sin  a  sin  fe  sin2  £C 

76.  cos  C  =  sin  A  sin  B  cos  c  —  cos  A  cos  B 

77.  sin  a  cos  c  =  sin  c  cos  a  cos  B  +  sin  b  cos  C 

78.  sin  a  cot  c  =  cos  <z  cos  B  +  sin  B  cot  C 

79.  sin  a  cos  B  =  sin  c  cos  b  —  sin  b  cos  c  cos  A 

Case  i.  Given  the  three  sides,  a,  b,  c. 
To  find  one  of  the  angles. 

80.  rirfiA  =  rin(*7\)'!n<*-c) 

sin  6  sin  c 

or, 

„  sin  k  sin  (&  —  a) 

81.  cos2|A  =  -  r-n-A  -  ' 

sin  o  sm  c 

82.  *. 


Case  ii.  Given  the  three  angles,  A,  B,  C 
To  find  one  of  the  sides. 


83-. 


., 
sin  B  sin  C 


400 


APPENDIX. 


or, 


84. 

85.   K  = 


=  cos(K  -  B)  cos  (K  -  C) 
sin  B  sin  C 


Case  in.  Given  two  sides  a  and  6,  and  the  included  angle  C. 
1°.  To  find  the  two  other  angles  A  and  B. 


86. 


87.   tan  J(A  -  B)  =cot  1C. 
2°.   To  find  the  third  side  c. 


or, 


T    .  i        i    • 

Napier's  Analogiea 


88. 


or  equa.  73. 

Case  iv.  Given  two  angles  A  and  B,  <wd  the  adjacent  side  c. 
1°.  To  find  the  other  two  sides,  a  and  b. 


89.   tan  £  (a  +  b)  =  tan 


cosi(A-B) 


90.    tan  J  («-&)=  tan  Jc. 


sn 


-  B) 


T    .    , 

Napier's  Analogies. 
P 


sm 


2°.  To  find  the  third  angle  C. 
-B). 


91. 


sin  5  (a  +  6) 


sn 


or, 


(a—  6) 

cos  i  (a-f  b) 


"  cos  \  (a — b) 
or  equa.  73. 

Case  v.  Given  two  sides  a,  5,  and  an  opposite  angle  A. 

To  find  the  other  opposite  angle  B ;  take  equation  73,  or  the 
proportion ;  sines  of  the  angles  are  as  sines  of  the  opposite  sides. 
(For  the  methods  of  determining  the  remaining  angle  and  side,  see 
page  402,  Case  3.) 

Case  vi.  Given  two  angles  A,  B,  and  an  opposite  side  a. 
To  md  the  other  opposite  side  b ;  sines  of  the  angle  are  proper- 


RESOLUTION  OF  SPHERICAL  TRIANGLES.  401 

tional  to  the  sines  of  the  opposite  sides.     (For  the  methods  of  de- 
termining the  remaining  side  and  angle,  see  page  402,  Case  4.) 

OTHER  METHODS  OF  RESOLVING  OBLIQUE-ANGLED  SPHERICAL 
TRIANGLES.* 

Except  when  three  sides  or  three  angles  are  given,  the  data 
always  include  an  angle  A,  and  the  adjacent  side  b,  besides  a  third 
part.  The  required  parts  in  the  different  cases  may  be  found  by 
the  following  formulae,  and  formula  73. 

92.   tan  m  =  tan  b  cos  A  93.   cot  n  =  tan  A  cos  b 

94.  c  =  m  +  m'  95.         C=n-fn' 

cos  a  _  cos  m1  cos  A  _  sin  n 

cos  b      cos  m  '    cos  B      sin/j' 

tan  A  _  sin  m'  tan  a  _  cos  n 

tan  B      sin  m  tan  b      cos  n' 

100.    sin  k  =  sin  A  sin  b. 

From  the  angle  C  (Fig.  120)  a  perpendicular  CD  is  let  fall  upon 
the  opposite  side  c,  which  divides  the  Fig.  120. 

triangle  into  two  right-angled  trian- 
gles, that  are  resolved  separately.  In 
the  one,  A  CD,  A  and  b  are  known, 
and  it  is  easy  to  find  the  other  parts, 
which,  joined  to  the  third  given  part, 
serve  to  resolve  the  second  right-an- 
gled triangle  BCD,  and  determine  the 
unknown  quantity  required,  m,  m' 
denote  the  two  segments  of  the  base ;  n,  n'  the  two  parts  of  the 
angle  C  ;  and  k  the  perpendicular  arc  CD. 

It  must  be  observed,  that  if  the  perpendicular  CD  fell  without 
the  triangle,  m  and  m',  n  and  n'  would  have  contrary  signs  ;  this 
happens  when  the  angles  A  and  B  at  the  base  are  of  different  kinds 
(the  one  ZL,  the  other  >90°).  When  it  is  not  known  whether  this 
circumstance  has  place  or  not,  the  problem  is  susceptible  of  two 
solutions. 

The  detail  of  the  different  cases  is  as  follows  :  the  data  are  A. 
b,  and  another  arc  or  angle. 

Case  1.  Given  two  sides  and  the  included  angle ;  or  b,  c,  A. 

Equation  92  makes  known  m,  94  m',  which  may  be  negative, 
(what  the  calculation  shows,)  96  a,  98  B,  and  equation  73,  (page 
399,)  C,  which  is  known  in  kind. 

Case  2.   Given  two  angles  and  the  adjacent  side;  or  A,  C,  b. 

Equation  93  makes  known  ft,  95  n',  which  may  be  negative 
(what  the  calculation  shows,)  97  B,  99  a ;  finally,  equation  78 
(page  399)  gives  c,  which  is  known  in  kind. 

*  Francceur's  Practical  Astronomy. 
26 


402  APPENDIX. 

Case  3.   Given  two  sides  and  an  opposite  angle;  cr  b,  a,  A. 

Equation  92  gives  m,  96  m't  94  c,  98  and  73  B  and  C ; 

or  else,  93  gives  n,  99  n',  95  C,  97  and  73  B  and  c. 

This  problem  admits  in  general  of  two  solutions.  In  effect,  the 
arc  m'  or  angle  n'  being  given  by  its  cos.,  may  have  either  the 
sign  -f-  or  — ;  there  are  then  two  values  for  c,  and  also  for  C.  m' 
and  n'  enter  into  equations  97  and  98  by  their  sines,  whence  result 
therefore  also  two  values  of  B. 

Case  4.   Given  two  angles,  and  an  opposite  side;  or  A,  B,  b. 

Equation  92  gives  m,  98  m',  94  c,  96  a,  and  equation  73  makes 
known  C  ; 

or  else  93  gives  n,  97  n',  95  C,  99  and  73  a  and  c. 

There  are  also  two  solutions  in  this  case  ;  for,  m1  or  n'  is  given 
by  a  sin.,  and  therefore  two  supplementary  arcs  satisfy  the  ques- 
tion. Thus  c  in  94,  and  a  in  96,  receive  two  values ;  same  for 
C  in  95,  and  a  in  99,  &c. 

Instead  of  solving  the  two  right-angled  triangles,  into  which  the 
oblique-angled  triangle  is  divided,  by  equations  92  to  99,  we  may 
employ  Napier's  rules,  from  which  these  equations  have  been  ob- 
tained. 

Isosceles  Triangles. 

When  the  triangle  is  isosceles,  B  =  C,  b  =  c,  the  perpendicular 
arc  must  be  let  fall  from  the  vertex  A,  and  the  equations  furnished 
by  Napier's  rules,  become  very  simple.  We  find 

101.  sin  i  a  =  sin  |  A  sin  b 

102.  tan  \  a  —  tan  b  cos  B 

103.  cos  b     =  cot  B  cot  i  A 

104.  cos  5  A  =  cos  5  a  sin  B     • 

The  knowledge  of  two  of  the  four  elements  A,  B,  a,  b,  which 
form  the  isosceles  triangle,  is  sufficient  for  the  determination  of  the 
two  others. 


INVESTIGATION  OF  ASTRONOMICAL  FORMULAE. 

Formula  for  the  Parallax  in  Right  Ascension  and  Declination, 
and  in  Longitude  and  Latitude.  (See  Article    93,  page  65.) 

Fig- 121  Let  s  (Fig.  121)  be  the  true  place 

of  a  star  seen  from  the  centre  of  the 
earth,  5' the  apparent  place,  seen  from 
a  point  on  the  surface  of  which  z  is 
the  zenith,  the  latitude  being  /.  The 
displacement  ss'  =  p  is  the  parallax 
in  altitude,  which  takes  effect  in  the  vertical  circle  zs' ;  p  is  the 


PARALLAX  IN  RIGHT  ASCENSION  AND  DECLINATION.  4Q3 

pole  ;  the  hour  angle  zps  =  q  is  changed  into  zps',  and  sps'  =  o 
is  the  variation  of  the  hour  angle,  or  the  parallax  in  right  ascen 
sion  ;  the  polar  distance  ps  =  d  is  changed  into  ps'  ;  the  differ- 
ence 5  of  these  arcs  is  the  parallax  in  declination  or  of  polar  dis- 
tance* We  have,  (For.  73,  p.  399,) 

sin  s1  :  s'mps  (d)  :  :  sin  sps'  (a)  :  sin  ssr  (p), 

sin  zps'  (q  -|-a)  :  sin  zs'  (Z)  :  :  sin  s'  :  sinpz  (90°—  I). 

Multiplying,  term  by  term,  we  obtain 

sin  sf  sin  (q  +  a)  :  sin  d  sin  Z  :  :  sin  a  sin  s'  :  sin  p  cos  I  : 

sin  p  cos  I   .    . 

whence,  sin  a  =   .    r    .  —  —  sm  (q  +  a). 

sm  a  sin  Z 

Or,  substituting  for  p  its  value  given  by  equa.  (8,)  p.  62,  and 
replacing  H  by  P, 

sin  P  cos  /  .     ,          N          ... 
sm  a  =  -  :  —  -  —  sin  (q  +  a)  .  .  .  (A). 
sin  ct 

This  equation  makes  known  a  when  the  apparent  hour  angle 
zps  —  q  -\-  a,  seen  from  the  earth's  surface,  is  given  ;  but  if  we 
know  the  true  hour  angle  zps  =  g,  seen  from  the  centre  of  the 
earth,  developing  sin  (q  +  a),  (For.  15,  p.  395),  and  putting 
sin  P  cos  I  _ 

sin  d 

sin  a  =  m  (sin  q  cos  a  +  sin  a  cos  q), 
or,  dividing  by  sin  a, 

1  =  m  (sin  q  cot  a  -f  cos  q)  ; 
whence,  by  transformation, 


.  „   .  ,  .    x 

tan  a  =  -  -  -  —  =  m  sin  q  +  m  sin  <7  cos  <7  (very  nearly.) 
1  —  m  cos  q 


Restoring  the  value  of  m, 

.        .    /sin  P  cos  Z\2    . 
sin  q  +  I  -  :  —  j  —  I    sin  q 
\      sm  d     / 


sin  P  cos  /   . 

tan  a  = ^— -= —  sin  q  +  { : — ; —  )    sin  q  cos 

sm  a 


Putting  the  arc  a  in  place  of  its  tangent,  and  P  in  place  of  sin  P, 
and  expressing  these  arcs  in  seconds,  (For.  47,  p.  397,)  there 
results, 

P  cos  Z   .  /P  cos/\2    . 

a  =  —  :  —  —  sin  q  +  I  —  :  —  3-  I    sin  q  cos  q  sm  1"  ...  (B). 
smd  \  smd  / 

The  parallax  in  declination  (5)  is  the  difference  of  the  arcs  ps 
(=  d)  and  ps'  (=d  +  5.)  Let  zs  =  z,  and  zs'  =  Z.  The  trian- 
gles zps  and  zps'  give  (For.  74  and  73), 

cos  d  —  sin  Z  cos  z      cos  (c?  +  5)  —  sin  /  cos  Z 

1°.   COS  »Z5  =  -  j—.  -  =  --  /-T-.  —  =  -  , 

cos  /  sin  z  cos  /  sm  Z 

*  Francceuf's  Uranography,  p.  418. 


404:  APPENDIX. 

sin  d  sin  7       sin  (d  +  5)  sm  (7  +  a) 

2°.    Sin  0ZS  =  -  :  -  =  --  :  --  ij  - 

sm  z  sm  /j 


.        „       cos  d  sin  Z  —  sm  /  cos  z  sin  Z  ,     .    ,        „ 

cos  (a  +  0)  = = r  sin  /  cos  Z 

v  sin  % 


From  the  first  equation  we  derive 
cos  d  sin  Z  —  sin 
sin  z 
_  cos  6?  sin  Z  —  sin  /  (cos  z  sin  Z  —  sin  z  cos  Z) 

sin  z 
_  cos  d  sin  Z  —  sin  I  sin  (Z  —  z) 

sin  z 
or,  (equ.  8,  p.  62,) 

sinZ  ,  •    -D    •    ?x 

=  —.  —  (cos  d  —  sm  P  sm  /)  : 
sm  z  v 

from  the  second, 

sinZ  _  sin  (d  +  $)   sin  (q  -f-  a). 

sin  z  sin  d  sin  ^ 

substituting, 

,         ~       sin  (d  +  <5)    sin  (</  -f  a)  ,        .          . 

cos  (d?  +  <5)  =  —  ^  —  -i—2  .  -  V  -  E  (cos  a  —  BUI  P  sin  A 
sin  a  sm  <? 

cos  (d  +  6)  _  sin  (</  +  a)  /cos  d       sin  P  sin  l\ 
sin  (d  +  <$)  sin  ^       Vsin  d  sin  d     / 

A  ,  7  ,  n       sin  (^  +  «)  /         ,      sin  P  sin  A         /r,. 

COt  ((/  +  5)  =  -  &  -  -  I  COt  d  --  :  -  y—  1  .  .  .  (C). 

sm  q       \  sm  d     / 

-T,  sin  P  sin  / 

Put  tan  x  =  -  :  —  _.  —  ; 
sm  d 

then,  cot  (<*+*)=  !^m-1'"—  (cot  rf  -  tan  a?) 

_sin  (^  +  «)  /cose?      sin  a?\ 
sin  ^      Vsin  ^     cos  a;/ 

_  sin  (^  +  a)   cos  rf  cos  a:  —  sin  d  sin  a: 
sin  q  sin  fi?  cos  x 

_sin  (g  +  «)  cos  (<Z  +  a;)  n. 

—  .  --  :  --  j  -  .    .    .   I  U  )  » 

sm  ^  sm  a  cos  a; 

The  apparent  polar  distance  (d  +  5)  being  computed  by  either 
of  the  formulae  (C)  and  (D),  we  have  8  =  (d  +  8)  —  d. 

Formulas  may  be  obtained  that  will  give  the  parallax  in  declina- 
tion without  first  finding  the  apparent  declination,  (except  approx- 
imately.) 

From  equa.  (C)  we  obtain 

sin  P  sin  I  _  sin  q  cot  (d  +  3) 

flint/  sin  (q  4  a)     ' 


PARALLAX  IN  RIGHT  ASCENSION  AND  DECLINATION.          405 

and  we  also  have 

*/j  i  j;\      cos  d     cos(d  +  <5)  sin  8 

cot  d  —  cot  (d  +  (5)  =  -  —  --  .    ,  .  ,     '  =  .     ,   . 
sin  d      sin  (</  4-  o)      sin  d  sin 

the  sum  of  these  equationo  gives 

sin  P  sin  /  _  \.^(^_         sm  3  _  \  j  sin 

J 


sind  sm(q  +  a.)         sin  d  sin  (d  +  <5)* 

v  sin  </       _  sin(y  -f  a)  —  sin  q 

i\  OW.  1    —  ~    ~      •        :       •  --  :  —         - 

sm  (q  +  «)  &m  (q  +  a) 

_  2  sin  $  a  cos  (q  +  £  a)  _  sin  a  cos  (7  -f  £  a) 

—  :     7       :       r  --  :  —  ~,  -  :  -  x  --  ;  —  (  T  Of.  22,    I  3  1 

sm  (^  +  a)  sm  (q  +  a)  cos  £  a 

cos  (<7  +  £  a)  sin  P  cos  I  . 

,  by  equa.  (A). 


sin  d  sin  (d  + 
or,  sin  5  =  sin  P  sin  I  sin  (d  -f  5)  — 

cos  (d  +  5)  cos  (q  +  i  a)  sin  P  cos  /       /T,X 
-  :  --  .  .  .  (rj) 

cos^a 

=  sin  P  sin  I  [sin  (rf  +  5)  —  tan  y  cos  (d  +  5)1 

.  .  cot  /  cos  (^  +  i  «) 

making  tany=  -   ^Y  --  -; 

cos^a 

.    .      sin  P  sin  I   .  t 

whence,          sin  d  =  --  sin  (rf  +  <5  —  y)  .  .  .  (F). 
cosy 

To  facilitate  the  calculation,  the  sines  of  5  and  P  in  eqs.  (E) 
and  (F),  may  be  replaced  by  the  arcs. 

To  obtain  an  expression  for  the  parallax  in  declination  in  terms 
of  the  true  declination,  develope  sin  (d  +  ^  —  y)  in  equation  (F) 
which  gives 

.    .      sin  P  sin  /  r  .  n  .  _      . 

sm  o  =  -  -  -  [sin  (d  •*{-  o)  cos  y—  sm  y  cos  (a  +  d)]  ; 

developing  sin  (d  +  5)  and  cos  (d  +  <*)>  and  reducing,  we  have 

.    r     sin  P 

sm  5  = 

dividing  by  cos  5, 


.    r     sin  P  sin  I  r  .    ,  ,       v  ,,       ». 

sm  5  =  --  [sm  (d—  y)  cos  8  +  cos  (J  —  y)  sm 


tan  5  =  SmSn    [sin  (d—  y)  +  cos  (d-y)  tan 


i06  APPENDIX, 

sin  P  sin  I    . 

sin  (d—  y) 

cosy 
whence  tan  6=         sinPsin/- 

1 cos  (d—y) 

cosy 


sin  P  sin  I 


.    ,j       N   .  /sin  P  sin  A2 

sm  (d—y)  + 1  - )  ; 

V     cos  y     / 


_       .    ,  /Psin/Vsinl"   .    ft/,       .         .  ,_  . 
—?/)  +  1  --  )   -  ^  —  sm2(d—  y)    .  .  .  (G.) 
V  cos     /       2 


cos  y 

sin  (d  —  y)  cos  (d—y)  (very  nearly  ;) 

or,  replacing  tan  5  and  sin  P  by  6  and  P,  expressing  these  arcs  in 
seconds,  (For.  47,  p.  397),  and  reducing  by  For.  13,  p.  395, 

Psin/    .    ,_       .    ,  /Psin/sinl" 
<$=  -  sm  (£?—?/)  +  1  -- 
cos  #  V  cos  y 

If  the  place  of  a  body  be  referred  to  the  ecliptic,  similar  formu 
lae  will  give  the  parallax  in  latitude  and  longitude,  but  as  the 
ecliptic  and  its  pole  are  continually  in  motion  by  virtue  of  the  di 
urnal  rotation  of  the  heavens,  it  is  necessary,  in  order  to  be  able  to 
determine  the  parallax  in  longitude  at  any  given  instant,  to  know 
the  situation  of  the  ecliptic  at  the  same  instant. 

This  is  ascertained  by  finding  the  situation  of  the  point  of  the 
ecliptic  90°  distant  from  the  points  in  which  it  cuts  the  horizon, 
and  which  are  respectively  just  rising  and  setting,  called  the  Non- 
agesimal  Degree,  or  the  Nonagesimal. 

Fig.  122.  Let  K  (Fig.  122)  be  the 

pole  of  the  ecliptic  jfe,  p  the 
pole  of  the  equator  fa  ;/is 
the  vernal  equinox,  the  ori- 
gin of  longitudes  and  of 
right  ascensions;  Jibs  is  the 
eastern  horizon,  b  the  hor 
oscope,  or  the  point  of  the 
eclipticwhich  is  just  rising; 
pz  =  90°  —  /  (the  latitude 
of  given  place)  ;  Kp  =  u  the  obliquity  of  the  ecliptic.  The  circle 
Kznv  is  at  the  same  time  perpendicular  at  n  to  the  ecliptic  /&,  and 
at  v  to  the  horizon  hb  ;  it  is  a  circle  of  latitude  and  a  vertical  cir- 
cle, since  it  passes  through  the  pole  K  and  the  zenith  z  :  b  is  90° 
from  all  the  points  of  the  circle  Knv  ;  zn  is  the  latitude  of  the  ze- 
nith, fn  its  longitude  ;  the  point  n  is  the  nonagesimal,  since  bn  = 
90°  ;  nv  is  the  altitude  of  this  point,  and  the  complement  of  zn  ; 
nv  measures  the  inclination  of  the  ecliptic  to  the  horizon  at  the 
given  instant,  or  the  angle  b,  so  that  b  =  nv  =  Kz  ;  thusfn  =  N 
the  longitude  of  the  nonagesimal,  and  nv  =  h  the  altitude  of  the 
nonagesimal,  designate  the  situation  of  this  point,  and  conse- 
quently ascertain  the  position  of  the  ecliptic  and  its  pole  at  the 
moment  of  observation.* 

*  Francceur's  Uranography,  p.  421 


FIG.  117. 


LONGITUDE  AND  ALTITUDE  OF  THE  NONAGE  SIMAL.     407 

The  points  m  and  d  are  those  of  the  equator  and  ecliptic  which 
are  on  the  meridian  ;  the  arc  fm,  in  time,  is  the  sidereal  time  s. 
which  is  known  ;  the  arc/z  =  90°,  since  the  plane  Kpi,  passing 
through  the  poles  K  and  p,  is  at  the  same  time  perpenolicular  to 
the  ecliptic  and  to  the  equator  ;  the  arc  mi  —fi  —fm  =  90°  —  s  ; 
then  the  angle  zpK  =  I80°—zpi  =  ISO0—  mi  =  90°+  s* 

Now,  in  the  spherical  triangle  pKz  we  know  the  sides  Kp  =  «, 
zp  =  90°—  /  =  H,  and  the  included  angle  zpK  —  90°  -f  s  ;  and 
may  therefore  find  Kz  —  h  the  altitude  of  the  nonage  simal,  and  the 
angle  pKz  —  nc  —fa~fn  =  90°—  N  =  complement  of  the  longi- 
tude N  of  the  nonagesimal.  Let  S  =  sum  of  the  angles  Kzp  and 
zKp,  then,  (For.  86,  page  400,) 


but, 

tan  £S  =—  tan  (180°—  iS),  and  tan  i  (90°—  s)  =—  tan  £  (5—90°)  ; 

substituting,  and  denoting  (180°  —  |S)  by  E,  we  have 


Again,  letD  =  zKp—Kzp,  then,  (For.  87,) 

tan  ID  -^Ig-"*    cot  i  (90-+*); 

sin  5  (H  +  w) 

whence,  by  transforming  as  above,  and  denoting  (180°  —  ^D)  by  F 
we  have 


-£-  tan  *  (<- 

sin  k  (H-h  w) 
Now,  iS  +  W  =pKz  =  90°—  N  ; 

whence,  N  =  90°-  ($S  +  iD), 

or, 

N  =  360°  +90°-  (iS  +  iD)  =  180°-iS  +180°-|D  +90°; 

consequently,  N  =  E  +  F  +  90°  .  .  .  (J), 

rejecting  360°  when  the  sum  exceeds  that  number. 

Next,  for.  the  altitude  of  the  nonagesimal,  we  have,  (For.  88,) 


cos  f 


N  and  h  being  known,  to  obtain  the  formula  for  the  parallax 
in  longitude  and  latitude,  we  have  only  to  replace  in  the  formulae 

*  Francoeur's  Uranography,  p.  421. 


408  APPENDIX. 

for  the  parallax  in  right  ascension  and  declination,  the  altitude  /  of 
the  pole  of  the  equator  by  that  90°—  h  of  the  pole  K  of  the  eclip 
tic,  and  the  distance  im  of  the  star  s  from  the  meridian  by  the  dis- 
tance nc  to  the  vertical  through  the  nonagesimal.  Let  us  change 
then  in  formula  (A),  (B),  (C),  (D),  (E),  (F),  and  (G),  /  into 
90°—  ft,  and  q  into  fc  —  fn  =  L—  N,  L  being  the  longitude  fc  of 
the  star  s.  Besides,  d  will  become  the  distance  sK  to  the  pole  of 
the  ecliptic,  or  complement  of  the  latitude  X  =  sc.  Making  these 
substitutions,  and  denoting  the  parallax  in  longitude  by  n,  and  the 
parallax  in  latitude  by  *r,  we  obtain  in  terms  of  the  apparent  longi- 
tude and  latitude, 

sin  P  sin  h      .    ,  T  . 

sinn  =  —  .—  i  —  (smL—  N  +  II)  .  .  .  (L), 

x      sin  (L—  N  +  n)/  sin  P  cos  h\  /1l/rx 

cot  (d  +  «)  =  -  ^-7=  —  =H—  *  I  cold  --  r—  5  —  1   ...  (M), 
sm(L-N)      V  smd      / 

sin  P  cos  h 


sin(L—  N)  sin 

sin  if  —  sin  P  cos  h  sin  (d  +  «r)  — 

cos  (d  +  if)  cos  (L—  N  +  ^n)  sin  P  sin  h 


sin  P  cos  h  .     ...  .  .  v 

sin  *  =  --  sin  (a  +  «—y)  .  .  .  (R); 

and  in  terms  of  the  true  longitude  and  latitude, 


P  cos  h   .    . 

*  — sin  (d  —  y]  + 

cosy 


_  tanAcos(L--N-Hn) 
cos  in 

To  facilitate  the  computation,  sin  n,  sin  «,  and  sin  P,  in  formu. 
lae  (L),  (P),  and  (R),  maybe  replaced  by  the  arcs  themselves. 

The  distance  d  of  the  star  from  the  pole  of  the  ecliptic  enters 
into  these  formulae  in  place  of  the  latitude  X. 

To  find  the  apparent  distance  d',  we  have 
d'  =  d  +  « 


409 

for  the  apparent  latitude  X', 

X'=X— ••; 
for  the  apparent  longitude  L', 

L'=L  +  n. 

The  logarithmic  formulas  given  on  page  352,  were  derived  from 
equations  (L),  (O),  and  (P),  and  the  logarithmic  formula  on  page 
353  from  equa.  (O). 

To  determine  now  the  effect  of  parallax  upon  the  apparent  di- 
ameter of  the  moon. 

Let  ACB  (Fig.  71,  p.  163)  represent  the  moon,  and  E  the  sta- 
tion of  an  observer ;  also  let  R  =  apparent  semi-diameter  of  the 
moon,  and  D  =  its  distance.  The  triangle  AES  gives 

AT-«C.         AS  AS 

sin  AES  =  ~— r    or   sin  R  =  -TT-. 
.no  D 

At  any  other  distance  D'  we  should  have  for  the  apparent  semi- 
diameter  R7, 

AS 
sin  R'  =  -yrj- ; 

sin  R'     D 

whence,  -r— -  =  — . 

sin  R      D' 

Thus,  if  R'  =  moon's  apparent  semi-diameter  to  an  observer  at  the 
earth's  surface,  as  at  O  (Fig.  34,  p.  61),  R  =  the  same  as  it  would 
be  seen  from  the  centre  C,  and  S  represents  the  situation  of  the 
moon, 

sin  R'  =  CS  =  sin  ZOS  =  sin  Z 
sinR  ~OS~sinZCS~~sin*  * 
But  we  have,  (see  page  404,) 

sinZ  _ (sine?  -f<5)    si 


sin  z  sin  d  sin  q 

•r,  in  terms  of  the  apparent  longitude  and  latitude,  (see  page  408,) 
sin  Z  _sm(d  +  «)  sin  (L  —  N  +  n) 
sin  z          sin  d  sin  (L  —  N) 


Hence,      sin  R'  =  sin  Rsi"  (<*  +  *)  *>  (L  -  N  +  n)  _  _ 

sin  d  sin  (L  —  N) 


Aberration  in  Longitude  and  Latitude,  and  in  Right  Ascension 
and  Declination*  (See  Art.  100,  page  70.) 

Aberration  is  caused  by  the  motion  of  light  in  conjunction  with 
the  motion  of  the  earth.  Light  comes  to  us  from  the  sun  in  8llu 
17"- .8,  during  which  time  the  earth  describes  an  arc  a  =  20'  .44, 

*  Francceur's  Uranography,  p.  442,  &c. 


410 


APPENDIX. 


of  its  orbit  pbdin  (Fig.  1 23,)  supposed  circular  :  p  is  the  place  of 
the  earth.     Let  us  take  any  plane  whatsoever,  which  we  will  cal] 
Fig.  123.  relative,  passing  through  the  star  and  the 

sun,  and  let  dd'  be  the  intersection  oi  this 
plane  and  the  ecliptic,  with  which  it  makes 
an  angle  k  :  let  us  seek  the  quantity  9  by 
which  the  aberration  displaces  the  star  in 
the  direction  perpendicular  to  this  plane. 
The  question  is  to  project  on  to  a  line  per- 
pendicular to  the  relative  plane,  the  small 
constant  arc  a  which  the  earth  describes, 
this  being  the  quantity  that  the  star  is  dis- 
placed from  its  line  of  direction  in  a  direction  parallel  to  the  line 
of  the  earth's  motion,  (see  Art.  196  of  the  text :)  this  projection 
is  9,  variable  according  to  the  position  of  the  relative  plane  in  rela- 
tion to  which  it  is  estimated.  The  velocity  along  the  tangent  at 
p,  makes  with  ph  an  angle  Q  =pch  ==  the  arc  pd' ;  a  cos  6  is  then 
the  projection  of  this  velocity  on  the  line  ph.  The  angle  of  our 
two  planes  being  k,  this  projection  will  be  reduced  to  a  cos  0  sin 
k,  when  it  is  taken  perpendicularly  to  the  relative  plane.  Thus, 

9  =  a  sin  k  cos  6  . .  .  (V). 

The  aberration  displaces  the  star  from  the  relative  plane  by  this 
quantity  9,  k  designating  the  inclination  of  this  plane  to  the  eclip- 
tic, and  6  the  arc  pd',  reckoned  from  p  the  place  of  the  earth  to  d' 
the  point  of  intersection  of  these  two  planes.  Let  us  give  to  the 
relative  plane  the  positions  which  are  met  with  in  applications. 

Let  us  suppose  at  first  that  k  =  90°,  or  sin  k  —  1  ;  the  relative 
plane  will  then  be  perpendicular  to  the  ecliptic.  Let  n  be  the  ver- 
nal equinox  ;  we  have  pd'  =  np  —  nd' ;  np  is  the  longitude  of  the 
earth,  or  180°  +  that  O  of  the  sun  ;  nd'  is  the  longitude  /  of  the 
star ;  whence 

9  =  —  a  cos  (O  —  /). 

Fig.  124.  Now,  let  M  (Fig.  124)  be  the  true  place 

^  .~  K  of  the  star,  M'  the  star  as  displaced  by 
aberration,  KM  is  the  circle  of  true  lati- 
tude, KM'  the  circle  of  apparent  latitude, 
and  MM'  =  9  :  this  arc  has  its  centre  C 
on  the  axis  which  passes  through  the  pole 
K  of  the  ecliptic  ;  the  longitude  of  the 
star  is  then  altered  by  the  part  OO'  of  the 
ecliptic  comprised  between  these  two 
planes  ;  and  since  OO'  is  to  the  arc  MM' 
as  the  radius  1  is  to  the  radius  CM  =  sin  KM  =  cos  latitude  X  of 
the  star,  we  have 

aberr.  in  long.  = cos  (O  —  1)  .  .  .  (W). 

cos  X 

If  the  relative  plane  iskc,  (Fig.  125,)  perpendicular  to  the  circle 


ABERRATION  IN  RIGHT  ASCENSION  AND  DECLINATION.        411 

of  latitude  Kcd,  the  aberration  <p 
j  erpendicularly  to  it,  will  be  the 
aberration  in  latitude.     Let  kd  be 
the  ecliptic,  and  o  the  earth  ;  the 
angle  A-  is  measured  by  the  arc  cd 
=  X  ;  the  arc  ok  =  6  =  O  —  •  long. 
of  7i  ;  and  as  kd  —  90°,  long,   of 
point  k  =  I  —  90°  :  substituting  in  equation  (V),  we  find 
aberr.  in  lat.  =  —  a  sin  X  sin  (O  —  1)  •  •  •  (X). 
These  aberrations  of  the  star  produce  a  small  apparent  orbit, 
which  is  confounded  with  its  projection  on  the  tangent  plane  to 
the  celestial  sphere.     Let  us  suppose  the  orbit  to  be  referred  to 
two  co-ordinate  axes  passing  through  the  true  place  of  the  star  and 
lying  in  the  tangent  plane,  of  which  one  is  parallel  to  the  plane  of 
the  ecliptic,  and  the  other  perpendicular  to  this,  or  tangent  to  the 

circle  of  latitude  at  the  star  ;  and  let  -  •  =  aberr.  in  long.,  and 

cos  X 

y  =  aberr.  in  lat.  ;  y  will  be  the  ordinate,  and  x  (the  aberr.  in  long., 
reduced  to  the  parallel  through  the  star)  the  abscissa  :  we  have 

x  a 

-  -  =  --  —  cos  (O  —  0, 
cos  X  cos  X 

y  =  —  a  sin  X  sin  (  O  —  I)  ; 

M 

or,  —  =  —  cos  (  O  —  I), 

=  —  sin(O-Z). 


asinX 

Squaring  the  last  two  equations,  and  adding  them  together,  O  dis- 
appears, and  we  find 

!/2  +  x2sin2X=a2sin2X  .  .  .  (Y), 

whatever  may  be  the  place  of  the  earth.  Such  is  the  equation  of 
the  apparent  orbit,  which,  as  we  perceive,  is  an  ellipse  of  which 
the  semi-axes  are  a  and  a  sin  X,  and  whose  centre  is  the  true  place 
of  the  star.  When  the  star  is  at  the  pole  of  the  ecliptic,  X  =  90°, 
and  the  ellipse  becomes  a  circle  of  which  the  radius  is  a.  When 
X  =  0,  this  ellipse  is  reduced  to  an  arc  2a  of  the  ecliptic. 

To  find  the  aberration  in  right  ascension,  the  relative  plane  must 
be  perpendicular  to  the  equator.  Let  kc  be  the  equator,  (Fig.  125,) 
p  its  pole,  psd  the  relative  plane,  which  is  the  circle  of  declination 
of  the  star  s  ;  kd  the  ecliptic,  o  the  earth,  k  the  vernal  equinox, 
kc  =  R,  sc  =  D.  Aberration  carries  the  star  s  out  of  the  plane 
pcd  a  distance  o>,  which  it  is  the  question  to  determine.  Equa. 
(V)  is  here 

<p  =  a  sin  d  cos  do  =  a  sin  d  cos  (kd  —  ko) 
=  a  sin  d  (cos  kd  cos  ko  +  sin  kd  sin  ko) 
=  a  sin  d  cos  kd  cos  ko  +  a  sin  d  sin  kd  sin  ko 


412  APPENDIX. 

but  ko  --  long,  of  earth  =  180°  -f  O  ;  we  have  also  the  angle  k  =• 
the  obliquity  w  of  the  ecliptic,  and  the  right-angled  spherical  trian 
gle  kcd  gives,  by  Napier's  rules, 

cot  kd  =  cot  R  cos  w,  sin  d  sin  kd  =  sin  R. 
The  1st  equa.  multiplied  by  the  2d,  gives 

sin  d  cos  kd  =  cos  R  cos  w, 
whence         <p  =  —  a  (cos  R  cos  w  cos  O  +  sh  R  sin  O). 

The  displacement  from  M  to  M'(Fig.  124)  conducts,  as  before, 
to  the  division  of  (p  by  cos  D,  to  have  the  conesponding  arc  of  the 
equator  :  thus  the  aberration  in  right  ascension  is, 

u  =  —  a  sin  R  sec  D  sin  O  —  a  cos  w  cos  R  sec  D  cos  O  (Z). 

Taking  the  relative  plane  perpendicular  to  the  circle  of  declina- 
tion, we  find  for  the  aberration  in  declination, 

v  =  —  a  sin  D  cos  R  sin  O  —  a  cos  w  (tan  w  cos  D   —  sin  R  sin  D) 
cos  O  .  .  .  (a). 

These  formulas  may  easily  be  adapted  to  logarithmic  computa- 
tion: 

In  formula  (Z)  let  a  sin  R  sec  D  =  A,  and  a  cos  w  cos  R  sec 
D=B;  then, 

u  =  —  A(sinO-f--^-cosO)  .  .  .  (Z'). 
B       a  cos  w  cos  R  sec  D 

Put  tan  9  =  -r-  =  -  :  —  =:  --  ^r  -  =  COS  W  COt  R  .  .  .  (6) 

A  a  sin  R  sec  D 

and  we  shall  have 

,  sin  <p 
u  —  —  A  (sin  O  -\  --  -  cos  O) 

COS(p 

.  sin  O  cos  <p  +  sin  <p  cos  © 
cos  <p 

A 

=  —  -  sin  (O  +  9). 

COS  (p 

Restoring  the  value  of  A,  and  taking  -  ^  for  sec  D,  we  obtain 


COS 

asinR 


- 


The  auxiliary  arc  (p  is  given  by  equation  (b)  ;  it  must  be  substi- 
tuted in  equation  (c),  with  its  sign,  and  we  then  obta  n  u.  Tan 
(p,  and  the  co-efficient  of  sin  (  O  +  <p)  are  constant,  for  the  same  star, 
for  a  long  period  of  time,  since  these  quantities  vary  very  slowly 
with  w  and  the  precession.  Moreover,  the  co-efficient  of  sin 
(O+<p)  is  the  maximum  value  of  w,  since  it  answers  to  sin 
(O  +  9)  =  1  ^  Thus  we  shall  be  able  to  caLulate  in  advance,  foi 


NUTATION  IN  RIGHT  ASCENSION  AND  DECLINATION.  413 

any  designated  star,  the  values  of  <p  and  of  the  maximum  of  the  aber- 
ration in  right  ascension,  or  of  the  logarithm  of  this  maximum. 

The  results  of  these  calculations  for  50  principal  stars  are  given 
in  Table  XCI,  columns  entitled  M  and  9. 

If  in  equation  (a)  we  make  a  sin  D  cos  R  =  A;,  and  a  cos  w 
(tan  w  cos  D—  sin  R  sin  D)  =  B',  we  shall  have  the  equation 

B' 
v  =—  A'  (sin  O  +  -£7  cos  O), 

in  which  A'  and  B'  are  constants.  This  equation  is  of  the  same 
form  with  equa.(Z').  We  therefore  have,  in  the  same  manner  as 
for  the  right  ascension, 

_  B'  _  a  cos  w  (tan  w  cos  D  —  sin  R  sin  D) 
~A'~  a  sin  D  cos  R 

a  sin  w  cos  D  —  a  cos  w  sin  R  sin  D 
a  sin  D  cos  R 

sin  w  cot  D 

=  -  IF:  --  cos  w  tan  K,  .  .  .  (d\ 
cosR 

A'  a  sin  D  cos  R 

v=  --  rsin(O+d)  =  —  --  7  --  x 
cos  0  cos  & 


.  .  .  (e). 

&  is  given  by  equation  (d),  and  being  substituted  in  equation  (e), 
we  shall  have  v.  &  and  the  co-efficient  of  sin  (O  -H)  are  constant 
for  the  same  star,  and  we  can  therefore  calculate  in  advance  the 
value  of  this  arc,  and  of  the  co-efficient,  which  is  the  maximum 
of  the  aberration  in  declination.  Columns  entitled  6  and  N,  Table 
XCI,  contain  the  quantities  6.  and  the  logarithms  of  the  maxima  of 
the  aberration  in  declination  for  50  principal  stars. 

For  convenience  in  calculation,  the  angles  9,  d,  and  the  maxima, 
M,  N,  in  Table  XCI,  have  been  rendered  positive  in  all  cases. 
This  has  been  accomplished  by  adding  12s-  to  9  and  6  whenever 
the  calculation  conducted  to  a  negative  value,  and  by  adding  6s-  to 
O  +  9,  or  O  +  d,  whenever  the  co-efficient  had  the  sign  —  ,  (this 
sign  being  changed  to  +  ;)  in  this  manner  the  sign  of  each  of  the 
two  factors  is  changed,  which  does  not  alter  the  sign  of  the  pro 
duct. 

Formula  for  the  Nutation  in  Right  Ascension  and  Declination* 
(See  Article  124,  p.  90.) 

In  deriving  these  formulae,  we  must  begin  with  borrowing  cer- 
tain results  established  by  Physical  Astronomy.  It  has  been 
proved,  in  confirmation  of  Bradley's  conjectures,  that  the  phenom- 
ena of  nutation  are  explicable  on  the  hypothesis  of  the  pole  of  the 
earth  describing  around  its  mean  place  (that  place  which,  see  page 

*  Woodhouse's  Astronomy,  p.  357,  &c. 


414 


APPENDIX. 


87,  it  would  hold  in  the  small  circle  described  around  the  pole  of 
the  ecliptic,  were  there  no  inequality  of  precession)  an  ellipse,  in 
a  period  equal  to  the  revolution  of  the  moon's  nodes.  The  major 
axis  of  this  ellipse  is  situated  in  the  solstitial  colure  and  equal  to 
18".50  ;  it  bears  that  proportion  to  the  minor  axis  (such  are  the 
results  of  theory)  which  the  cosine  of  the  obliquity  bears  to  the 
cosine  of  twice  the  obliquity  :  consequently,  the  minor  axis  will  be 
13".77. 

Let  CdA.  (Fig.  126)  represent  such  an  ellipse,  P  being  the  mean 
place  of  the  pole,  K  the  pole  of  the  ecliptic.     CDOA  is  a  circle 

Fig.  126. 


described  with  the  centre  P  and  radius  CP.  VL  is  the  ecliptic, 
Vw  the  equator,  KPL  the  solstitial  colure.  In  order  to  determine 
the  true  place  of  the  pole,  take  the  angle  APO  equal  to  the  retro- 
gradation  of  the  moon's  ascending  node  from  V  :  draw  Oi  perpen- 
dicular to  PA,  and  the  point  in  the  ellipse,  through  which  Oi 
passes,  is  the  mie  place  of  the  pole.  This  construction  being  ad- 
mitted, the  nuiations  in  right  ascensi6n  and  north  polar  distance 
may,  Pp  being  very  small,  be  thus  easily  computed. 

Nutation  in  North  Polar  Distance. 

Nutation  in  N.  P.  D.  =  Ptf—  prf.  =  Pr  =  Pp  cospPrf,  nearly, 
=  P/>cos(APp-t-AP<r) 
=  Pp  cos  (APp  +  R  —  90°) 


R  denoting  the  right  ascension. 


NUTATION  IN  RIGHT  ASCENSION  AND  DECLINATION.  415 

Nutation  in  Right  Ascension. 

The  right  ascension  of  the  star  <f  is,  by  the  effect  of  nutation, 
changed  from  Vw  into  Vts.     Now, 

Vts  =  V'v  +  Vw  +  ts,  nearly, 
whence,  Vw;  —  ^Tfts  =  —  Vv  —  ts 

=  -  VV  cos  V  V'v  -  Pp  sin  Pprf 


in  which  expression  V'w  (=  VV  cos  VV'r)  is,  as  in  the  case  of  pre- 
cession", common  to  all  stars. 

In  order  to  reduce  farther  the  above  expression,  we  have 
pPa  =  APp  +  APrf  =  APp  +  R  -  90°, 


whence,          —  V'v  —  ts  =  —  Pp  sin  APp  cot  w 

-  Pp  sin  (APp  +  R  -  90°)  cot  N.  P.  D. 
=  —  Pp  sin  APp  cot  w  +  Pp  cos  (APp  +  R)  cot  <5, 

<5  representing  the  north  polar  distance,  and  u  the  obliquity  of  the 
ecliptic. 

But  these  forms  are  not  convenient  for  computation.     In  order 
to  render  them  convenient,  we  must,  from  the  properties  of  the  el- 
lipse, deduce  the  values  of  Pp,  and  of  the  tangent  of  APp,  and 
then  substitute  such  values  in  the  above  expressions  :  thus, 
Pp_  __  sec  APp       cos  APO  _  cos(128—  Q)  _     cos  Q 
PO  ~~  sec  APO  ~  cos  APp         cos  APp          cos  APp' 
&  designating  the  longitude  of  the  moon's  ascending  node  ; 

T,        PO  cos  & 

whence  Pp  =  -  --  —  . 

cos  APp 

tan  AP^  _    pi  _  Pd  __  Pd 
ta^APO~Oi~PD~~PO; 

hence,  tan  APp  =  p^  tan  APO  =  p^  tan  (12«-  -  fl) 

Pd 


Now  substitute,  and  there  will  result 

The  Nutation  in  North  Polar  Distance 

=  FO  co*  Q  (sin  APp  cos  R  +  cos  APp  sin  R) 
cos  APp 

=  PO  (tan  APp  cos  R  cos  ft  +  cos  ft  sin  R) 

=  —  Pd  cos  R  sin  ft  +  PO  cos  ft  sin  R 

=  -  6".887  cos  R  sin  ft  +  9r.250  cos  ft  sin  R  (/) 


416  APPENDIX. 

which  is  the  difference,  as  far  as  nutation  is  concerned,  between 
the  mean  and  apparent  north  polar  distance.  The  apparent  north 
polar  distance,  therefore,  must  bo  had  by  adding  the  preceding 
quantity,  with  its  sign  changed,  to  the  mean. 

Nutation  in  right  ascension  =  Pd  sin  &  cot  w 
+  PO  cos  &  cos  R  cot  <5  +  Pd  sin  &  sin  R  cot  $, 
which,  as  far  as  nutation  is  concerned,  is  the  difference  of  the  mean 
and  apparent  right  ascensions  :  and,  consequently,  the  above  ex- 
pi  ession  must  be  subtracted  from  the  mean,  in  order  to  obtain  the 
apparent  right  ascension ;  or,  which  is  the  same,  must  be^  added 
after  a  negative  sign  has  been  prefixed  ;  in  which  case,  we  have, 
substituting  for  PO,  Pd  their  numerical  values, 

The  Nutation  in  Right  Ascension 

—  —  6".887  sin  &  cot  w 
— 9".250  cos  Q,  cos  R  cot  8— 6". 887  sin  &  sin  R  cot  <5  .  .  .  (g). 

Formulae  (/)  and  (g)  are  of  the  same  form  with  (Z)  and  (a)  for 
the  aberrations  in  right  ascension  and  declination,  and  therefore 
formulae  may  be  derived  from  them  similar  to  (c)  and  (e),  adapted 
to  logarithmic  computation.  The  quantities  corresponding  to  <p, 
M,  d,  N,  have  been  calculated  for  the  stars  in  the  catalogue  of 
Table  XC,  and  inserted  in  Table  XCI,  in  the  columns  entitled 
<p',  M',  V,  N'. 

The  Solar  Nutation  arises  from  like  causes  as  the  Lunar,  and 
admits  of  similar  formulae.  As  an  ellipse,  made  the  locus  of  the 
true  place  of  the  pole,  served  to  exhibit  the  effects  of  the  lunar 
nutation,  so  an  ellipse,  of  different,  and  much  smaller  dimensions, 
may  be  made  to  represent  the  path  which  the  true  pole  of  the 
equator  would,  by  reason  of  the  sun's  inequality  of  force  in  caus- 
ing precession,  describe  about  the  mean  place  of  the  pole.  Thus, 
in  Figure  130,  the  ellipse  A.dC  will  serve  to  represent  the  locus 
of  the  pole,  when  AP  =  0".545,  Pd  =  0".500,  and  APO,  instead 
of  being  =  £,  is  equal  to  2  0,  or  twice  the  sun's  longitude, 
taken  in  the  order  of  the  signs  ;  the  equations,  therefore,  for  the 
solar  nutation  in  north  polar  distance,  and  right  ascension,  analo- 
gous to  eqs./and  g  will  be 

The  Solar  Nutation  in  North  Polar  Distance 
=  -  0".500  cos  R  sin  2  0  +  0".545  sin  R  cos  2  ©  .  .  .  (h). 

The  Solar  Nutation  in  Right  Ascension 
=  —  0".500  sin  2  ©  cot  w 
-  0".545  cos  2  ©  cos  R  cot  8  —  0".500  sin  2  ©  sin  R  cot  5  . .  (i). 

If  the  apparent  place  of  a  star  should  be  required  with  great 
precision,  it  would  be  necessary  to  compute  the  solar  nutations 
from  these  formulae,  and  apply  them  as  corrections  to  the  mean 


417 

right  ascension  and  declination.  The  calculation  would  be  per- 
formed after  the  same  manner  as  for  the  lunar  nutation ;  but  it  is 
much  abridged  by  remarking  that  the  form  of  the  equations  is  the 
same  as  that  of  the  equations  for  the  lunar  nutation,  and  that  the 
co-efficients  are  very  nearly  the  0.075  "of  those  of  the  latter  equa- 
tions. Thus  we  can  make  use  of  the  same  arcs  <p',  <9',  and  log. 
maxima,  M',  N',  repeat  the  calculation  for  the  lunar  nutation, 
taking  2  O  instead  of  &>,  and  multiply  the  nutations  in  right  ascen- 
sion and  declination  thus  obtained  by  0.075.  The  results  will  be 
the  solar  nutations  required.  (See  rrob.  XX.) 


F '  rrmul&for  computing  the  effects  of  the  Oblateness  of  the  Earth's 
Surface  upon  the  Apparent  Zenith  Distance  and  Azimuth  of  a 
Star* 

From  the  centre  of  the  earth,  an  observer  would  see  a  star  at  I, 
(Fig.  127,)  and  would  have  V  for  his 
zenith  :  from  the  surface  his  zenith  is 
Z,  and  he  sees  this  star  at  B  ;  IB  =p 
is  the  parallax  in  altitude ;  the  azi- 
p  mdth  VZI  is  changed  into  VZB.  If 
for  a  given  time,  we  wish  to  calculate 
the  apparent  zenith  distance  BZ,  and 
the  apparent  azimuth  VZB,  we  have 
first  to  resolve  the  spherical  triangle  IZP,  in  which  we  know  the 
two  sides  ZP  =  co-latitude  and  IP  =  co-declination,  and  the  in- 
cluded hour  angle  P  ;  the  azimuth  VZI  (=  A),  and  the  arc  IZ 
(=  n)  will  thus  be  known.  But  from  the  earth's  surface,  the  star 
is  seen  at  B  :  the  azimuth  VZB  =  VZI  +IZB  =  A  +  « ;  the  zenith 
distance  BZ  =  n  -}-p,  since,  VZ  (=  i)  being  very  small,  we  have 
sensibly  IB  +  IZ  =  BZ.  By  reason  of  the  want  of  sphericity  of 
the  earth,  parallax  then  increases  the  true  azimuth  and  zenith 
distance  of  a  star  by  small  quantities,  a  and  p,  which  it  is  neces- 
sary to  calculate.  In  the  triangle  VIZ  we  have 

cos  IV  =  cos  i  cos  n  -f-  sin  i  sin  n  cos  A  =  cos  n  -f-  k  sin  n  ^ 

making  cos  i  =  1 ,  sin  i  =  i,  and  i  cos  A  =  #.     Now,  k  L  z,  and 
d  fortiori  cos  k  =  1 ,  sin  k  =  k  ;  whence 

cos  IV  =  cos  n  cos  k  +  sin  n  sin  k  =  cos  (n  —  k)* 
and  IV  =7i  —  k  =n  —  i  cos  A. 

Thus  we  correct  the  calculated  arc  n  by  the  quantity  —  i  cos 
A,  to  have 

IV  =  z  —  n  —  i  cos  A  ...  (j). 
If  this  value  of  z  be  introduced  into  equation  (#),  page  422,  we 

*  Francoeur's  Uranography,  p.  426,  &c. 


418 


APPENDIX. 


shall  have  p,  and  thence  the  apparent  zenith  distance  Z  =  n 
=  EZ. 

Afterwards,  to  obtain  IZB  =  a,  or  the  parallax  in  azimuth,  the 
triangles  ZBV,  ZBI  give 

sin  ZBV  _  sin  (A  +  a)        sin  ZBV  _  sin  a 

sin  i          sin  (z  -{-p) '          sin  n          sinp  ' 
whence,  by  equating  the  values  of  sin  ZBV, 

sin  n  sin  a  _  sin  i  sin  ( A  +  «)  _ 

sin  p  sin  (z  +  p) 

substituting  for  sin  p  its  value  sin  H  sin  (z  +_p)  =  sin  H  sin  Z, 
(equa.  8,  page  51,)  and  reducing,  we  have 

sin  a        _  sin  (A  +  a) 
sin  H  sin  i  sin  n 

and  as  i  is  very  small,  sin  i  sin  (A  +  a)  does  not  differ  sensibly 
from  i  sin  A,  and  we  thus  have  in  seconds,  (For.  47,  page  397,) 

Hi  sin  A  sin  1"  7N 

a  = : .  .  .  (k). 

sin  n 


Solution  of  Kepler's  Problem,  by  which  a  Body's  Place  is  found 
in  an  Elliptical  Orbit*    (See  Art.  199,  p.  127.) 

Let  APB  (Fig.  128)  be  an  ellipse,  E  the  focus  occupied  by  the 
sun,  round  which  P  the  earth  or  any  other  planet  is  supposed  to 
revolve.  Let  the  time  and  planet's  motion  be  dated  from  the  ap- 

Fig.  128. 

M 


Woodhouse's  Astronomy,  p.  457,  &c. 


SOLUTION  OF  KEPLER'S  PROBLEM.  419 

cutting  off,  from  the  whole  ellipse,  an  area  AEP  bearing  the  same 
proportion  to  the  area  of  the  ellipse  which  the  given  time  bears  to 
the  periodic  time. 

There  are  some  technical  terms  used  in  this  problem  which  we 
will  now  explain. 

Let  a  circle  AMB  be  described  on  AB  as  its  diameter,  and  sup- 
pose a  point  to  describe  this  circle  uniformly,  and  the  whole  of  it 
in  the  same  time  as  the  planet  describes  the  ellipse  ;  let  also  t  de- 
note the  time  elapsed  during  P's  motion  from  A  to  P  ;  then  if  AM  = 

-  r—  -.   x  2  AMB,  M  will  be  the  place  of  the  point  that  moves 

uniformly,  while  P  is  that  of  the  planet;  the  angle  ACM  is 
called  the  Mean  Anomaly,  and  the  angle  AEP  is  called  the  True 
Anomaly. 

Hence,  since  the  time  (t)  being  given,  the  angle  ACM  can  al- 
ways be  immediately  found,  (see  Art.  198,  p.  127,)  we  may  vary 
the  enunciation  of  Kepler's  problem,  and  state  its  object  to  be  the 
finding  of  the  true  anomaly  in  terms  of  the  mean. 

Besides  the  mean  and  true  anomalies,  there  is  a  third  called  the 
Eccentric  Anomaly  ,  which  is  expounded  by  the  angle  DC  A,  and 
which  is  always  to  be  found  (geometrically)  by  producing  the  ordi- 
nate  NP  of  the  ellipse  to  the  circumference  of  the  circle.  This 
eccentric  anomaly  has  been  devised  by  mathematicians  for  the 
purposes  of  expediting  calculation.  It  holds  a  mean  place  between 
the  two  other  anomalies,  and  mathematically  connects  them.  There 
is  one  equation  by  which  the  mean  anomaly  is  expressed  in  terms 
of  the  eccentric  ;  and  another  equation  by  which  the  true  anomaly 
is  expressed  in  terms  of  the  eccentric. 

We  will  now  deduce  the  two  equations  by  which  the  eccentric 
is  expressed,  respectively,  in  terms  of  the  true  and  mean  anomalies. 
Let  t   =  time  of  describing,  AP, 

P  =  periodic  time  in  the  ellipse, 

a   =CA, 

ae  =  EC, 

v  =  L  PEA, 

u  =  L  DC  A  ;  (whence,  ET,  perpendicular  to  DT,  =  EC 
x  sin  u,) 

P   =PE, 

*   =  3.14159,  &c.  ; 

then,  by  Kepler's  law  of  the  equable  description  of  areas, 
,_Px     area  PEA  area  DEA  __  P 

t  —  -  r   X   -  F~Tr  -  —  A    X  -  :  —  -  —  --  ~-  (  IJJ^O  -f"  JJUA) 

area  01  ellip.  area  circle       var 

ET.DC   ,  AD.DC 
~          —  — 


P  Pi 

—  (e  sin  u  +  u)  :  hence,  if  we  put  —  =  -, 


4:20  APPENDIX. 

we  have 

nt  =  e  sin  u  +  u  .  .  .  (/), 

an  equation  connecting  the  mean  anomaly  nt,  and  the  eccentric  u. 
In  order  to  find  the  other  equation,  that  subsists  between  the 
true  and  eccentric  anomaly,  we  must  investigate,  and  equate,  two 
values  of  the  radius-vector  p,  or  EP. 

First  value  of  p,  in  terms  of  v  the  true  anomaly, 

==«(l_:1e3>B 
l  —  e  cost;  ' 

Second,  in  terms  of  u  the  eccentric  anomaly, 

p   =  a   (1  +  e  cos  u)  .  .  .  (2). 
For,  p2  =  EN2  +  PN2 

=  EN2  +  DN2x(l  -e2) 

=    (ae  4-  a  cos  w)2  +  fla  sin2  u  (1  —  e2) 

=  az\#  +  2e  cos  u  +  cos2  u\  +  a2  (  1  —  e'J)  sin2  u 

=  a2    1  +  2e  cos  u 


Hence,  extracting  the  square  root, 

p  =  a  (1  +e  cos  u). 
Equating  the  expressions  (1),  (2),  we  have 

(1  —  e2)  =  (I  —  e  cos  v)  (1  -\-  e  cos  u),  whence, 

e  -f  cos  % 

cos  ?;  =  -—  :  --  ,  an  expression  lor  v  in  terms  of  u  : 
1  +  e  cos  % 

but,  in  order  to  obtain  a  formula  fitted  to  logarithmic  computation, 
we  must,  find  an  expression  for  tan  -  :  now,  (see  For.  12,  p.  397,) 

tan  -  =  x  /  (l  ~cosv\  -    ,  /  ((l-*)(l-cosu)\ 
'2       v    Vl+cosu/        V    \(l  +  e)  (I  +  cos  u)J 


These  two  expressions  (Z)  and  (m),  that  is, 
ft£  =  e  sin  M  + 


analytically  resolve  the  problem,  and,  from  such  expressions,  by 
certain  formulae  belonging  to  the  higher  branches  of  analysis,  may 
v  be  expressed  in  the  terms  of  a  series  involving  nt. 

Instead,  however,  of  this  exact  but  operose  and  abstruse  method 
of  solution,  we  shall  now  give  an  approximate  method  of  express- 
ing the  true  anomaly  in  terms  of  the  mean. 

MO  is  drawn  parallel  to  DC.     (1.)  Find  ihe  half  difference  of 


SOLUTION  OF  KEPLER'S  PROBLEM.  421 

the  angles  at  the  base  EM  of  the  triangle  ECM,  from  this  ex- 
pression, 

tan  i  (GEM  -  CME)  =  tan  \  (GEM  +  CME)  x  !ll-e, 

1  +  e 

in  which  GEM  +  CME  —  ACM,  the  mean  anomaly. 

(2.)  Find  GEM  by  adding  |  (GEM  +  CME)  and  \  (GEM  — 
CME)  and  use  this  angle  as  an  approximate  value  to  the  eccen- 
tric anomaly  DCA,  from  which,  however,  it  really  differs  by 
/  EMO. 

(3.)  Use  this  approximate  value  of  /DCA  =  /EOT  in  com- 
puting ET  which  equals  the  arc  DM  ;  for,  since  (see  p.  419), 

p 

t  =  --  -  x  DEA,  and  (the  body  being  supposed  to  revolve 
area  circle 

in  the  circle  ADM)  =  -  ?—  -  x  ACM,  area  AED  =  area  ACM, 
area  circle 

or,  area  DEC  +  area  ACD  =  area  DCM  +  area  ACD  ;  con- 
sequently the  area  DEC  =  the  area  DCM,  and,  expressing  their 
values. 

XDO  and  thug  ET  = 


Having  then  computed  ET  =  DM,  find  the  sine  of  the  resulting 
arc  DM,  which  sine  =  OT  ;  the  difference  of  the  arc  and  sine 
(ET  —  OT)  gives  EO. 

(4.)  Use  EO  in  computing  the  angle  EMO,  the  real  difference 
between  the  eccentric  anomaly  DCA  and  the  /MEG;  add  the 
computed  /  EMO  to  /MEG,''  in  order  to  obtain  /DCA.  The 
result,  however,  is  not  the  exact  value  of  /DCA,  since  /EMO 
has  been  computed  only  approximately;  that  is,  by  a  process 
which  commenced  by  assuming  /MEG  for  the  value  of  the 
/DCA. 

For  the  purpose  of  finding  the  eccentric  anomaly,  this  is  the 
entire  description  of  the  process,  which,  if  greater  accuracy  be 
required,  must  be  repeated;  that  is,  from  the  last  found  value  of 
/  DCA  =  /  EOT,  ET,  EO,  and  /EMO  must  be  again  computed. 

Formula  for  calculating  the  Parallax  in  Altitude  of  a  Heavenly  Body  from  its  True 
Ze-ilh  Distance.  (See  Art.  88,  p.  62.) 

In  the  actual  state  of  astronomy,  the  true  co-ordinates  of  the  places  of  the 
heavenly  bodies  are  generally  known,  or  may  be  obtained  by  computation  from 
the  results  of  observations  already  made,  and  from  these  there  is  often  occasion 
to  deduce  the  apparent  co-ordinates.  For  this  purpose  there  is  required  an  ex- 
pression for  the  parallax  hi  altitude  hi  terms  of  the  true  zenith  distance. 

If  we  make  Z=z+p  hi  equation  (8)  p.  62,  we  shall  have 

sin  p 
sin  p  =  sin  H  sin  .(z  +p\  or  sin  H  = 


whence, 

sin  p         sin  (z  +p)  +  sin  p 

=  - 


422 


APPENDIX. 


and 

Dividing,  • 
or, 


— sinH=l  — 


sin  p      _  sin  (z  +  p}— sin  p 
sin  (z+p)  ~        sin  (z+p) 


1  +sin  H  _sin  (z+p)  +  sinp  . 
1—  sin  H  ~sin  (z+p)—  sinjp' 


tang2  (45°+  *  H)  = 


(App.  For.  36,  29); 


tang  ^  z 
whence, 

tang (i  z+^)=tang  £  z  tang2  (45°  +  i  H). . .  .(a). 

This  equation  makes  known  £  z+p,  from  which  we  may  obtain^)  by  subtract- 
ing \  z. 

Formula  for  computing  the  Annual 
Variations  in  the  Right  Ascension  and  De- 
clination of  a  Heavenly  Body.  (See  Art. 
119,  p.  88.) 

Let  VLA  (Fig.  129)  be  the  ecliptic, 
K  its  pole,  PP'P"  the  circle  described 
by  the  mean  pole,  P  the  mean  pole, 
and  YQA  the  mean  equator  at  any 
given  time,  P'  the  mean  pole  and  V  Q'A' 
the  mean  equator  a  year  afterwards,  and 
s  a  star.  Draw  P'r  perpendicular  to  the 
declination  circle  Psa.  We  have 
an.  var.  in  dQC.=sa'—sa=Ps—P's=Pr; 
but  since  PPV  may  be  considered  as  a 
right-angled  plane  triangle, 
Pr=PP'  cos  P'Pr=PP'  sin  QPa (a). 

Regarding  KPP'  as  a  right-angled  isosceles  triangle,  we  obtain 

sin  KPP'  or  1  :  sin  KP' ::  sin  PKP'  :  sin  PP' ; 
whence, 

sin  PP'  =  sin  PKP'  sin  KP ,  or  PP'=PKP'  sin  KP'  (nearly), 
substituting  in  equation  (a),  there  results, 

Pr=PKP'  sin  KP'  sin  QPa. 

PKP'  =  50".24;  KP'  =  obliquity  of  the  ecliptic  =  a> ; 

QPa=VQ— Va=90°— R  (R  designating  the  right  ascension  of  the  star  s).    Thus, 
finally, 

an.  var.  in  dec.  =  50". 24  sin  w  cos  R (c). 

Next,  we  have 

an.  var.  hi  r.  asc.=Y'a'—  Va=V'a'— ml=Vm  +  la' (a1); 

but, 

V'm=VV  cos  W'm=50".24  cos  «; 
and  since  the  right-angled  triangles  sP'r  and  sla  are  similar, 

sin  sr  or  sin  sP'  (nearly)  :  sin  P'r ::  sin  sa'  :  sin  la' ; 
whence, 

sin  sa'  sin  sa' 

sin  la  =sm  P  r  ^rr^rj  or  la'  =  P'r— — 5—  (nearly). 


(&): 


The  triangle  PPV  gives  PV=PP'  sin  P'Pr=PP'  cos  QPa^PKP'  sin  KP'  coa 
QPa  (equa.  6);  and  sin  P'*=cos  sa.     Substitutmg,  we  obtain 


Id  =PKP'  sin  KP'  cos 


sin  KP'  cos  QPa  tang  sa'. 


HELIOCENTRIC   LONG.   AND  LAT. 


423 


Eeplacing  PKP',  KP',  and  QPa  by  their  values,  as  above,  and  taking  the  declina- 
tion sa  for  sa  and  denoting  it  by  I),  there  results, 

&a'=50".24  sin  w  sin  R  tang  D. 
Now,  substituting  in  equation  (d)  the  values  of  V'm,  and  ba',  we  have 

an.  var.  in  r.  asc.=50".24  cos  w  +  50".24  sin  e»  sin  B  tang  D (e) 

The  results  of  formulae  (c,  e,  )  are  to  be  used  with  their  algebraic  signs,  if  the 
reduction  is  from  an  earlier  to  a  later  epoch,  otherwise  with  the  contrary  signs. 
The  declination  is  always  to  be  considered  positive  if  North,  and  negative  if  South. 

V'm=50".24  cos  o>=50".24  cos  23°27'=46".0, 
is  the  annual  retrograde  motion  of  the  equinoctial  points  along  the  equator. 

Formulae,  for  computing  the  Heliocentric  Longitude  and  Latitude,  and  Radius-vector 
of  a  Planet,  from  its  Geocentric  Longitude  and  Latitude.  (Referred  to  in  Art  177, 
p.  119.) 

The  longitude  of  the  node  and  the  inclination  of  the  orbit  are  supposed  to  be 
known.  Let  NP  (Fig.  130)  be  part  of  the  orbit  of  a  planet,  SNC  the  plane  of  the 


FIG.  130. 


ecliptic,  N  the  ascending  node,  S  the  sun,  E  the  earth,  and  P  the  planet ;  also,  let 
PT  be  a  perpendicular  let  fall  from  P  upon  the  plane  of  the  ecliptic,  and  EV,  SV, 
the  direction  of  the  vernal  equinox.  Let  A  =  PEir  the  geocentric  latitude  of  the 
planet ;  Z  =  PSr  its  heliocentric  latitude ;  G  =  VE^r  its  geocentric  longitude ;  L  = 
VSrr  its  heliocentric  longitude ;  S  =  VES  the  longitude  of  the  sun ;  N  =  VSN  the 
heliocentric  longitude  of  the  node ;  I  =  PNC  the  inclination  of  the  orbit ;  r  =  SE 
the  radius-vector  of  the  earth ;  and  v  •=  SP  the  radius-vector  of  the  planet. 

The  point  *  is  called  the  reduced  place  of  the  planet,  and  S*  its  curtate  distance. 
All  the  angles  of  the  triangle  SE*  have  also  received  particular  appellations ;  S^E 
the  angle  subtended  at  the  reduced  place  of  the  planet  by  the  radius  of  the  earth's 
orbit,  is  called  the  Annual  Parallax,  SE-r  the  Elongation,  and  ES*  the  Commutation. 
Let  A  =  SrE,  E  =  SEr,  and  C  =  ESr.  Draw  Sr'  parallel  to  E*:  then  A  =  irgir'  = 
VSr—  VSr'  =VST  —  VET=L  —  G-;  E  =  VET  — VES  =  G  — S;  C  =  VSE  — 
VS-T  =  180°  +  VSE'  —  VS*  =  180°  +  VES  —  VSr  =  180°  +S  —  L  =  T— L 
(putting  T  =  180°  +  S). 

(1.)  For  the  latitude.— The  triangles  EP*-,  SP*,  give 

ET  tang  X  =  P*  =  Srr  tang  Z,  whence  tepg>  =  ?I ; 

tang  I      ET 


out, 


: :  sin  E  :  sin  C,  or,  j?I  =  ^l^J?; 
ET      sin  C 


424  APPENDIX. 

Bubstituting,  tangX^sinJE 

tang  I      sin  G 

whence,  tang  X  sin  0  =  tang  I  sin  E.  .  .  .(a). 

or,  tang  X  sin  (T  —  L)  =  tang  Z  sin  (G  —  S)  ____  (&). 

Again,  the  triangle  NPp  gives,  by  Napier's  first  rule, 

sin  Np  =  cot  PNp  tan  Pp,  or,  sin  (L  —  N)  =  cot  I  tan  I  ____  (c). 
Either  of  the  equations  (&)  and  (c)  will  give  the  value  of  Z,  when  the  longitude  L 
is  known. 

(2.)  For  the  Longitude.  —  If  we  substitute  in  equation  (&)  the  value  of  tang  Z,  given 
by  equation  (c),  and  replace  (G  —  S)  by  E,  we  have 

tang  A  sin  (T  —  L)  —  sin  (L  —  N)  tang  I  sin  E; 

but  T  —  L  =  (T  —  N)  —  (L  —  '  N)  =  D  —  (L  —  N),  (denoting  (T  —  N)  by  D)  ;  sub- 
Btituting,  and  designating  L  —  N  by  &, 

tang  A  sin  (D  —  JB)  =  sin  a;  tang  I  sin  E  ; 
whence, 

tang  X  sin  D  cos  cc  —  tang  X  cos  D  sin  a;  =  tang  I  sin  E  sin  a;, 
or,  tang  X  sin  D  —  tang  X  cos  D  tang  x  =  tang  I  sin  E  tang  x, 

which  gives 

tang  x  =  _  taagXjtoD  -----  ^ 

tang  X  cos  D  +  tang  I  sm  E 
Substituting  the  values  of  x,  D,  and  E,  we  have,  finally, 

tang  (L-N)  =  _  _  J^gJ^inJT  ~N)  ___        ^  ^ 

tang  X  cos  (T  —  N)  +  tang  I  sin  (G  —  S) 
As  N  is  known,  the  value  of  L  will  result  from  this  equation. 

The  co-ordinates  employed  to  fix  the  position  of  a  planet  in  the  plane  of  its 
orbit,  are  its  orbit  longitude  and  its  radius-vector,  both  of  which  result  from  the 
heliocentric  longitude  and  latitude,  the  longitude  of  the  node  and  the  inclination 
of  the  orbit  being  known. 

In  Fig.  130,  V'NP  represents  the  orbit  longitude,  and  SP  (  =  v)  the  radius-vec- 
tor, for  the  position  P.  Now,  the  triangle  PSn-  gives 


SP  =  _!-.,  or,  „=       .; 

COS  PS*  COS  I 


and  the  triangle  ESn-  gives 


sin  A          sin  A 
whence,  by  substitution, 

r  sin  E  r  sin  (G  —  S)  /«' 

•  •  •  n/  /• 


sin  A  cos  I      sin  (L  —  G)  cos  I 
The  orbit  longitude  L'  =  NP  +  long,  of  node. . .  .(g) : 
and  to  find  NP,  the  triangle  NPp  gives 

cos  PNp  =  cot  NP  tang  Np,  or  tang  NP  = 

cos  I 
and  Np  =  long,  of  planet  —  long,  of  node. 

Formula  for  computing  the  Geocentric  Longitude  and  Latitude  of  a  Planet  from  its 
Heliocentric  Longitude  and  Latitude  and  Radius-  Vector. 

Let  S  (Fig.  130)  be  the  sun,  E  the  earth,  P  the  planet,  *  its  reduced  place,  and  V 
the  vernal  equinox.  Denote  the  heliocentric  longitude  VSn-  by  L,  the  heliocentric 
latitude  PS*  by  Z,  and  the  radius- vector  SP  by  v ;  and  denote  the  geocentric  longi- 
tude by  G,  and  the  geocentric  latitude  by  X  Also  let  E  =  SE*  the  elongation  ;  C 
=  ES?r  the  commutation ;  A  =  Sn-E  the  annual  parallax  ;  and  r  —  SE  the  radius- 
vector  of  the  earth.  JSTow, 

VEn-  =  SE;r  +  YES, 

or  G  =  E  +  long,  of  sun. 


GEOCENTRIC  LONG.   AND  LAT.  425 

This  equation  will  make  known  the  geocentric  longitude  when  the  value  of  E  is 
found.  In  the  triangle  PSn-  the  side  ST  =  SP  cos  PSr  =  v  cos  L,  and  is  therefore 
known,  the  side  ES  is  given  by  the  elliptical  theory,  and  the  angle  C  may  be  de- 
rived from  the  following  equation:  C=  VSE  —  V&T  =  long,  of  earth  —  long,  of 
planet  ;  and  to  fiud  E  we  have,  by  Trigonometry, 

ES  +  ST  :  ES  —  ST  ::  tan  £  (E*S  +  SEir)  :  tan  £  (E^S  —  SET), 
or,  r  +  v  cos  I  :  r  —  v  cos  I  ::  tang  £  (A  +  E)  :  tang  £  (A  —  E)  ; 

whence, 

v  cos  Z 

f  -  i)  cos  /  f 

tang  ±  (A  —  E)  =  —^~^J  tang  £(A  +  E)  =  --  ^^  tang  *  (A  +  E). 


V  COS  I 

Let  tang  0  =  -  :  then, 


1  —  tang  9 
° 


or,  tang  £  (A  —  E)  =  tang  (45°  —  6)  tang  ±  (A  +  E).  .  .  .(a) 

But,  A  +  E  =  180°—  C,  andE=i  (A  +  E)  —  £  (A  —  E). 

Xext,  to  find  the  geocentric  latitude. 

ST  tang  I  =  PT  =  ET  tang  X 

whence, 


En-  -  tang  /  » 

ST       sin  E 
but,  ST  :  ET  ::  sin  E  :  sin  C,  or  |r  =  ^Q"  f 

sin  E       tang  X 
and  therefore  =' 

or 

"When  a  planet  is  in  conjunction  or  opposition,  the  sines  of  the  angles  of  elonga- 
tion and  commutation  are  each  nothing.  In  these  cases,  then,  the  geocentric  latitude 
cannot  be  found  by  the  preceding  formula;  it  may,  however,  be  easily  determined 
in  a  different  manner.  Suppose  the  planet  to  be  in  conjunction  at  P,  (Fig.  56,  p. 
120  ;)  then, 

PT  PT 


But  the  triangle  S?T  gives 

PT  ==  v  sin  I  and  ST  =  v  cos  Z,  and  ES  =  r  ; 

v  sin  I 
tang  X  =  T 


To  find  the  distance  of  the  planet  from  the  earth,  represent  the  distance  by  I)  ; 
then,  from  the  triangles  PT&  and  EPr,  we  have 

PT  =  EP  sin  PET  =  D  sin  X, 
and  PT  =  SP  sin  P&T  =  v  sin  I  • 


The  distance  of  a  planet  being  known,  its  horizontal  parallax  may  be  computed 
from  the  equation 

sin  H  =  —  ....  (e.)    (Art.  88). 


426  APPENDIX. 


CALCULATION  OF  AN  ECLIPSE   OF  THE  SUN.. 

(1).   Of  the  circumstances  of  the  general  eclipse. 

It  is  a  simple  inference  from  what  lias  been  established  in  Art.  333,  that  an 
eclipse  of  the  sun  will  begin  and  end  upon  the  earth,  at  the  tunes  before  and  after 
conjunction,  when  the  distance  of  the  centres  of  the  moon  and  sun  is  equal  to 
P— p  +  6  +  d;  that  the  total  eclipse  will  begin  and  end  when  this  distance  is  equal 
to  P— p— <5  +  d;  and  the  annular  eclipse  when  the  distance  is  equal  to  P— p  +  S 
— d. 

The  times  of  the  various  phases  of  the  general  eclipse  of  the  sun  may  be  obtained 
by  a  process  precisely  analogous  to  that  by  which  the  times  of  the  phases  of  an 
eclipse  of  the  moon  are  found.  Let  C  (Fig.  131)  be  the  centre  of  the  sun,  and  C' 
the  centre  of  the  moon,  at  the  time  of  conjunction.  We  may  suppose  the  sun  to 
remain  stationary  at  C,  if  we  attribute  to  the  moon  a  motion  equal  to  its  motion 
relative  to  the  sun  ;  for,  on  this  supposition,  the  distance  of  the  centres  of  the  two 
bodies  will,  at  any  given  period  during  the  eclipse,  be  the  same  as  that  which  ob- 
tains in  the  actual  state  of  the  case.  Let  N'C'L'  represent  the  orbit  that  would  be 
described  by  the  moon  if  it  had  such  a  motion,  which  is  called  the  Relative  Orbit 
Let  CM  be  drawn  perpendicular  to  it;  and  let  Cf  =  Gf  =  P — p  +  t  +  d,  and  C<7  — 
Gg'=  P — p  —  6  +  d,  or  P — p  +  A — d,  according  as  the  eclipse  is  total  or  annular. 
Then,  M  will  be  the  place  of  the  moon's  centre  at  the  middle  of  the  eclipse;/ and  f 
the  places  at  the  beginning  and  end  of  the  eclipse ;  and  g  and  g'  the  places  at  the 
beginning  and  end  of  the  total,  or  of  the  annular  eclipse.  We  shall  thus  have,  as 
in  eclipses  of  the  moon, 


tang  I  =      U    i  CM  =  *  cos  J:  C'M  —  ^  sin  I 
3600s.  >  sin  I  cos  I  ,  . 


Interval  from  con.  to  mid.    = 

Interval  from  middle  to  beginning  or  end 
3600s.  cos  I 


=     M-m     V&  +  A  cos  I)  (*'~A  cos  I)  ...  (b). 
Interval  for  total  eclipse 

3600s.  cos  I    /  __________ 

=  ~~MlT^r"y  (*"  +  A  COB  J)  (*"—  A  cos  J)  •  •  •  (c)- 

Interval  for  annular  eclipse 

3600s.  cos  I    ,  __  .  ______ 

=  ~~MfHUT"~     ^'"  +  A  cos  I)  (#"—  A  cos  I)  ...  (d). 

6  (V  —  A  cos  I) 
Quantity  =  -     —  j—    -  ...  (e). 

k"  =  P  —  p  —  3  +  d,k'"  =  P  —    +  i  —  d  . 


The  letters  A,  M,  m,  &c.,  represent  quantities  of  the  same  name  as  in  the  formula 
for  a  lunar  eclipse  ;  but  they  designate  the  values  of  these  quantities  at  the  time  of 


CALCULATION  OF  AN  ECLIPSE  OF  THE  SUN.  427 

conjunction,  instead  of  opposition.      These  values  are  in  practice  obtained  from 
tables  of  the  sun  and  moon,  as  in  a  lunar  eclipse. 

The  times  of  the  different  circumstances  of  a  general  eclipse  of  the  sun  may  also 
be  found  within  a  minute  or  two  of  the  truth,  by  construction,  in  a  precisely  similar 
manner  with  those  of  an  eclipse  of  the  moon  (330). 

(2  )   Of  the  phases  of  the  eclipse  at  a  particular  place. 

The  phase  of  the  eclipse,  which  obtains  at  any  instant  at  a  given  place,  is  indi- 
cated by  the  relation  between  the  apparent  distance  of  the  centres  of  the  sun  and 
moon,  and  the  sum,  or  difference,  of  their  apparent  semi-diameters;  and  the  calcula- 
tion of  the  time  of  any  given  phase  of  the  eclipse,  consists  in  the  calculation  of  the 
time  when  the  apparent  distance  of  the  centres  has  the  value  relative  to  the  sum  or 
difference  of  the  semi-diameters,  answering  to  the  given  phase.  Thus,  if  we  wish 
to  find  the  time  of  the  beginning  of  the  eclipse,  we  have  to  seek  the  time  when  the 
apparent  distance  of  the  centres  of  the  sun  arid  moon  first  becomes  equal  to  the 
sum  of  their  apparent  semi-diameters. 

The  calculation  of  the  different  phases  of  an  eclipse  of  the  sun,  for  a  particular 
place,  involves,  then,  the  determination  of  the  apparent  distance  of  the  centres  of 
the  sun  and  moon,  and  of  the  apparent  semi-diameters  of  the  two  bodies  at  certain 
stated  periods. 

The  true  semi-diameter  of  the  sun,  as  given  by  the  tables,  may  be  taken  for  the 
apparent  without  material  error.  For  the  method  of  computing  the  apparent  semi- 
diameter  of  the  moon,  for  any  given  time  and  place,  see  Problem  XVII. 

According  to  the  celebrated  astronomer  Dusejour,  in  order  to  make  the  observa- 
tions agree  with  theory,  it  is  necessary  to  diminish  the  sun's  semi-diameter,  as  it  is 
given  by  the  tables,  3".5.  This  circumstance  is  explained  by  supposing  that  the  ap- 
parent diameter  of  the  sun  is  amplified  by  reason  of  the  very  lively  impression  which 
its  light  makes  upon  the  eye.  This  amplification  is  called  Irradiation.  He  also 
thinks  that  the  semi-diameter  of  the  moon  ought  to  be  diminished  2",  to  make  allow- 
ance for  an  Inflexion  of  the  light  which  passes  near  the  border  of  this  luminary,  sup- 
posed to  be  produced  by  its  atmosphere.  It  must  be  observed,  however,  that  the 
astronomers  of  the  present  day  do  not  agree  either  as  to  the  necessity  or  the  amount 
of  the  diminutions  just  spoken  of. 

The  determination  of  the  apparent  distance  of  the  centres  of  the  sun  and  moon 
may  easily  be  accomplished,  as  will  be  shown  in  the  sequel,  when  the  apparent  longi- 
tude and  latitude  of  the  two  bodies  have  been  found.  Now,  the  true  longitude  of  the 
sun,  and  the  true  longitude  and  latitude  of  the  moon,  may  be  found  from  the  tables, 
(Probs.  IX.  and  XIV.);  and  from  these  the  apparent  longitudes  and  latitudes  may  be 
deduced  by  correcting  for  the  parallax.  But  the  customary  mode  of  proceeding  is  a 
little  different  from  this :  the  true  longitude  and  latitude  of  the  sun  are  employed  in- 
stead of  the  apparent,  and  the  parallax  of  the  sun  is  referred  to  the  moon  ;  that  is, 
the  difference  between  the  parallax  of  the  moon  and  that  of  the  sun  is,  by  fiction, 
taken  as  the  parallax  of  the  moon.  This  supposititious  parallax  is  called  the  moon's 
Relative  Parallax.  (See  Prob.  XVII.) 

We  will  first  show  how  to  find  the  approximate  times  of  the  different  phases  of  the 
eclipse.  Put  T  =  the  time  of  new  moon,  known  to  within  5  or  10  minutes.  (Prob. 
XXVII.)  For  the  time  T  calculate  by  the  tables  the  sun's  longitude,  hourly  motion, 
and  semi-diameter,  and  the  moon's  longitude,  latitude,  horizontal  parallax,  semi- 
diameter,  and  hourly  motions  in  longitude,  and  latitude.  Subtract  the  sun's  hori- 
zontal parallax  from  the  reduced  horizontal  parallax  of  the  moon,*  and  calculate 
the  apparent  lougitude  and  latitude,  and  the  apparent  semi-diameter  of  the  moon. 
From  a  comparison  of  the  apparent  longitude  of  the  moon  with  the  true  lon- 
gitude of  the  sun,  we  shall  know  whether  apparent  ecliptic  conjunction  occurs 
before  or  after  the  time  T.  Let  T'  denote  the  time  an  hour  earlier  or  later  than  the 
time  T,  according  as  the  apparent  conjunction  is  earlier  or  later.  With  the  sun  and 
moon's  longitudes,  the  moon's  latitude,  and  the  hourly  motions  in  longitude  and  lati- 
tude, at  the  time  T,  calculate  the  longitudes  and  the  moon's  latitude  for  the  time  T'; 
and  lor  this  time  also  calculate  the  moon's  apparent  longitude  and  latitude.  Take 
the  difference  between  the  apparent  longitude  of  the  moon  and  the  true  longitude  of 
the  sun  at  the  time  T,  and  it  will  be  the  apparent  distance  of  the  moon  from  the 
sun  in  longitude,  at  this  time.  Let  it  be  denoted  by  n.  Find,  in  like  manner,  the 
apparent  distance  of  the  moon  from  the  sun  in  longitude  at  the  time  T',  and  denote 

*  The  reduced  horizontal  parallax  of  the  moon  is  its  horizontal  parallax  as  re- 
duced from  the  equator  to  the  given  place.  (See  Prob.  XV.) 


I 


428 


APPENDIX. 


it  by  ft'.  In  the  same  manner  as  at  the  time  T,  we  find  whether  apparent  conjunc- 
tion occurs  before  or  after  the  time  T'.  If  it  occurs  between  the  times  T  and  T',  the 
sum  of  v,  and  ri,  otherwise  their  difference,  will  be  the  apparent  relative  motion  of  the 
sun  and  moon  in  longitude  in  the  interval  T' — T,  or  T — T' ;  from  which  the  relative 
hourly  motion  will  become  known.  The  difference  of  the  apparent  latitudes  of  the 
moon,  at  the  times  T  and  T',  will  make  known  the  apparent  relative  hourly  motion 
in  latitude.  With  the  relative  hourly  motion  in  longitude  and  the  difference  of  the 
apparent  longitudes  at  the  time  T,  find  by  simple  proportion  the  interval  between 
the  time  T  and  the  time  of  apparent  ecliptic  conjunction ;  and  then,  with  the  apparent 
latitude  of  the  moon  at  the  time  T  and  its  hourly  motion  in  latitude,  find  the  appar- 
ent latitude  at  the  time  of  apparent  conjunction  thus  determined.  Then,  knowing 
the  relative  hourly  motion  of  the  sun  and  moon  in  longitude  and  latitude,  together 
with  the  time  of  apparent  conjunction,  and  the  apparent  latitude  at  that  time,  and  re- 
garding the  apparent  relative  orbit  of  the  moon  as  a  right  line  (which  it  is  nearly), 
it  is  plain  that  the  time  of  beginning,  greatest  obscuration,  and  end,  as  well  as  the  quan- 
tity of  the  eclipse,  may  be  calculated  after  the  same  manner  as  in  the  general  eclipse  ; 
the  disc  of  thesuuausweringto  the  section  of  the  luminous  frustum  mentioned  in  Art. 

331,  and  the  apparent  elements 
answering  to  the  true.  Let  C 
(Fig.  132)  represent  the  centre 
of  the  sun  supposed  stationary, 
CC'  the  apparent  latitude  of  the 
moon  at  apparent  conjunction, 
N'C'  the  apparent  relative  orbit 
of  the  moon,  determined  by  its 
passing  through  the  point  C'  and 
making  a  determinate  angle  with 
the  ecliptic  N'F,  or  by  its  pass- 
ing through  the  situations  of  the 
moon  at  the  times  T  and  T'.  Also, 
let  RKFK'  be  the  border  of  the 
sun's  disc ;  /  /'  the  positions  of 
the  moon's  centre  at  the  begin- 
ning and  end  of  the  eclipse,  determined  by  describing  a  circle  around  C  as  a  centre, 
with  a  radius  equal  to  the  sum  of  the  apparent  semi-diameters  of  the  sun  and  moon; 
and  M  (the  foot  of  the  perpendicular  let  fall  from  C  upon  N'C')  its  position  at  the 
time  of  greatest  obscuration. 

If  the  eclipse  should  be  total  or  annular,  then  g,  g'  will  be  the  positions  of  the  moon's 
centre  at  the  beginning  and  end  of  the  total  or  annular  eclipse;  these  points  being 
determined  by  describing  a  circle  around  C  as  a  centre,  and  with  a  radius  equal  to 
the  difference  of  the  apparent  semi-diameter  of  the  sun  and  moon. 

The  results  will  be  a  closer  approximation  to  the  truth,  if  the  same  calculations 
that  are  made  for  the  time  T'  be  made  also  for  another  time  T". 

The  various  circumstances  of  the  eclipse  may  also  be  had  by  construction,  after 
the  same  manner  as  in  a  lunar  eclipse  (330). 

In  order  to  be  able  to  observe  the  beginning  or  end  of  a  solar  eclipse,  it  is  neces- 
sary to  know  the  position  of  the  point  on  the  sun's  limb  where  the  first  or  last  con- 
tact takes  place.  The  situation  of  these  points  is  designated  by  the  distance  on  the 
limb,  intercepted  between  them  and  the  highest  point  of  the  limb,  called  the  Vertex. 
The  contacts  wiil  take  place  at  the  points  t,  t',  (Fig.  132,)  on  the  lines  Of,  Of.  To 
find  the  position  of  the  vertex,  with  the  SUP'S  longitude  found  for  the  beginning  of 
the  eclipse,  calculate  the  angle  of  position  of  the  sun  at  that  time,  (see  Prob.  XIII.), 
and  lay  it  off  to  the  right  of  the  circle  of  latitude  CK,  when  the  sun's  longitude  is  be- 
tween 90°  and  270°,  and  to  the  left  when  the  longitude  is  less  than  90°  or  more  than 
270°.  iSuppose  CP  to  be  the  circle  of  declination  thus  determined.  Next,  let  Z 
(Fig.  6,  p.  13)  be  the  zenith,  P  the  elevated  pole,  and  S  the  sun ;  then  in  the  trian- 
gle ZPS  we  shall  know  ZP  the  co-latitude.  ZPS  the  hour  angle  of  the  sun,  and  we 
may  deduce  PS,  the  co-declinatiori  of  the  sun,  from  the  longitude  of  the  sun  as  de- 
rived from  the  tables  (equ.  24).  These  three  quantities  being  known,  ZSP,  the 
angle  made  by  the  vertical  through  the  sun  with  its  circle  of  declination,  may  be 
computed ;  and  being  laid  off  in  the  figure  to  the  right  or  left  of  CP  (Fig.  132),  ac- 
cording as  the  time  of  beginning  is  before  or  after  noon,  the  point  Z  or  Z',  ns  the  case 
may  be,  in  which  the  vertical  intersects  the  limb  RKK',  will  be  the  vertex,  and  the 


FIG.  132. 


CALCULATION  OF  AN  ECLIPSE   OF  THE  SUN.  429 

arc  Z/.  Z't.  on  the  limb,  will  ascertain  the  situation  of  t,  the  first  point  of  contact, 
with  respect  to  it. 

The  situation  of  the  last  point  of  contact  may  be  found  by  the  same  mode  of 
proceeding. 

Let  us  now  show  how  to  find  the  exact  times  of  the  beginning,  greatest  obscuration, 
and  end  of  the  eclipse,  the  approximate  times  being  known.  Let  B  designate 
the  approximate  time  of  beginning,  taken  to  the  nearest  minute.  Calculate  for 
the  time  B  by  means  of  the  tables,  the  suu's  longitude,  hourly  motion,  and  semi- 
diameter;  also  the  moon's  longitude,  latitude,  horizontal  parallax,  semi-diameter, 
and  hourly  motions  in  longitude  and  latitude.  Then,  making  use  of  the  relative 
parallax,  calculate  the  apparent  longitude,  latitude,  and  semi-diameter  of  the  moon. 
Subtract  the  apparent  longitude  of  the  moon  from  the  true  longitude  of  the  sun  ;  the 
difference  will  be  the  apparent  distance  of  the  moon  from  the  sun  in  longitude : 
let  it  be  denoted  by  a.  Denote  the  apparent  latitude  of  the  moon  by  A. 

Now,  let  EC  (Fig  133)  represent  an  arc  of  the  ecliptic, 
and  K  its  pole ;  and  let  S  be  the  situation  of  the  sun, 
and  M  the  apparent  situation  of  the  moon  at  the  time 
B.  Then  MS  is  the  apparent  distance  of  the  centres  of 
the  two  bodies  at  this  time.  Denote  it  by  A.  &rm=a, 
and  Mm  =.  A.  The  right-angled  triangle  MS/n,  being 
very  small,  may  be  considered  as  a  plane  triangle,  and 
we  therefore  have,  to  determine  A,  the  equation 

A*  =  a* +**.... (g)* 

Having  computed  the  value  of  A,  we  find,  by  com- 
paring it  with  the  sum  of  the  apparent  semi-diameters 
of  the  sun  and  moon,  whether  the  beginning  of  the 
eclipse  occurs  before  or  after  the  approximate  time  B.  E" 
Fix  upon  a  time  some  4  or  5  minutes  before  or  after  B, 
according  as  the  beginning  is  before  or  after,  and  call  it  FIG.  133. 

B'.     With  the  sun  and  moon's  longitudes,  the  moon's 

latitude,  and  the  hourly  motions  in  longitude  and  latitude,  at  the  time  B,  find  the 
longitudes  and  the  moon's  latitude  at  the  time  B',  and  compute  for  this  time  the 
apparent  longitude,  latitude,  and  semi-diameter  of  the  moon.  Subtract  the  apparent 
longitude  of  the  moon  from  the  true  longitude  of  the  sun,  and  we  shall  have  the 
apparent  distance  of  the  moon  from  the  sun  at  the  time  B'.  Take  the  difference 
between  this  and  the  same  distance  a  at  the  time  B,  and  we  shall  have  the  ap- 
parent relative  motion  of  the  sun  and  moon  in  longitude  during  the  interval  of  time 
between  B  and  B'.  Then  find,  by  simple  proportion,  the  apparent  relative  hourly 
motion  in  longitude,  and  denote  it  by  k.  Take  the  difference  between  the  apparent 
latitudes  of  the  moon  at  the  times  B  and  B',  and  it  will  be  the  apparent  relative 
motion  of  the  sun  and  moon  in  latitude,  in  the  interval ;  from  which  deduce  the 
apparent  relative  hourly  motion  in  latitude,  and  call  it  n.  Now,  put  t  =  the  inter- 
val between  the  approximate  and  true  times  of  the  beginning  of  the  eclipse,  and 
suppose  S  and  M  (Fig.  1 33)  to  be  the  situations  of  the  sun  and  moon  at  the  true 
time  of  beginning.  In  the  time  t,  the  apparent  relative  motions  in  longitude  and 
latitude  will  be,  respectively,  equal  to  Td  and  nt,  and  accordingly  we  shall  have 
Sm  =  a— Id,  MTTC  =\+ni. 

The  small  right-angled  triangle  SMra  may  be  considered  as  a  plane  triangle ;  the 
hypothenuse  SM=U/=the  sum  of  the  apparent  semi-diameters  of  the  sun  and  moon, 
minus  5".5  (p.  427).  We  have  then,  to  find  t,  the  equation 

(a -M)8+(*  +  *9*  =</'*i 
or,  developing  and  transposing, 

(n2  +k'2)  t2-2  (afc-Xn)  t  =  ti2-  («2  4  X2)  =  t//2—  A2  ; 

*  In  place  of  equation  (g)  the  following  equations  may  be  employed  in  logarith- 
mic computation : 

X  a     . 

tang  9  =  --'  A  = 

where  6  is  an  auxiliary  arc. 


430  APPENDIX. 

making  A=  t//2 —  A  *,  and  B  =  ale  —  Aw,  we  have  (w2  +  kz)  t*  —  2B£  =  A, 

B— -/B'2+A(/i2  +  P)  ,,v 

and  <  =  ' ^-^ ';Wffc 

The  negative  sign  must  be  prefixed  to  the  radical,  for,  if  we  suppose  A  to  be 
equal  to  zero,  t  must  be  equal  to  zero.  Multiplying  the  numerator  and  denomi- 
nator by  B-f  V  B2  +  A  (n*  +k*\  and  restoring  the  value  of  A  we  obtain 

3600s.  (A2—://2) 
(in  seconds),         (  =  ^=======  .  .  .  W. 

Although  this  equation  has  been  investigated  for  the  beginning  of  the  eclipse, 
it  is  plain  that  it  will  answer  equally  well  for  the  determination  of  the  other 
phases,  if  we  give  the  proper  values  and  signs  to  «//,  «,  A,  n,  and  Jc.  k  is  positive 
before  conjunction  and  negative,  after  it,  and  the  radical  quantity  is  negative  after 
conjunction ;  n  is  negative,  when  the  moon  appears  to  recede  from  the  north  pole 
of  the  ecliptic ;  A  has  the  sign  — ,  when  it  is  south ;  a  is  always  positive.* 

The  value  of  t  taken  with  its  sign  is  to  be  added  to  the  time  B. 

The  values  of  the  quantities  «,  A,  TO,  and  &,  are  found  for  the  other  phases  after 
the  same  manner  as  for  the  beginning. 

To  obtain  the  value  of  a<  at  the  time  of  greatest  obscuration,  find  the  relative 
motions  in  longitude  and  latitude  (k  and  n),  during  some  short  interval  near  the 
middle  of  the  eclipse,  which  is  the  approximate  time  of  greatest  obscuration ; 
then  compute  the  inclination  of  the  relative  orbit  by  the  equation 

n 
tang  I  =  -£-  .  .  .  (j). 

after  which  i//  will  result  from  the  equation 

<//  =  A  cos  I  .  .  .  (Jc). 

X  is  the  moon's  latitude  at  the  time  of  apparent  conjunction,  which  is  easily  cal- 
culated, by  means  of  the  values  of  k  and  n,  and  the  apparent  longitude  and  latitude 
of  the  moon,  found  for  some  instant  near  the  time  of  apparent  conjunction. 

For  the  beginning  and  end  of  the  total  eclipse,  we  have,  i//  =  appar.  semi-diam. 
of  moon  —  appar.  semi-diam.  of  sun  +  1".5 ;  and  for  the  beginning  and  end  of  the 
annular  eclipse,  -I  =  appar.  semi-diam.  of  sun  —  appar.  semi-diam.  of  moon  — 
1".5. 

If  the  value  of  »//,  given  by  equation  (&),  be  substituted  in  equation  («),  this 
equation  will  make  known  the  time  of  greatest  obscuration ;  but  this  may  be 
found  more  conveniently  by  a  different  process.     Let  NCF  (Fig.  134)  represent  a 
portion  of  the  ecliptic,  EML  a  portion  of 
the  relative  orbit  passed  over  about  the 
time  of  greatest  obscuration,  C  the  station- 
ary position  of  the  sun's  centre,  and  M  the 
place  of  the  moon's  centre  at  the  instant  of 
its  nearest  approach  to  0.     Also,  let  n  = 
CR  the  apparent  distance  of  the  moon  from 
the  sun  in  longitude  at  the  time  of  the  near- 
est approach  of  the  centres,  A'  =  RM  the 
moon's  apparent  latitude  at  the  same  time, 

k  =  MA;  the   apparent   relative  motion  in  FIG.  134. 

longitude  in  some  short  interval  about  this 
time,  and  n  =  kn  the  moon's  apparent  motion  in  latitude  during  the  same  interval. 
The  right-angled  triangles  M.nk  and  CMR  are  similar,  for  their  sides  are  respect- 
ively perpendicular  to  each  other ;  whence 

MJfc  :  MR  ::  kn  :  CE; 

kn  n 

a**  CR  =  MRF-  or,  a  =  A'  -^  .  .  .  (I). 

*  Developing  the  radical  in  equation  (ft),  and  neglecting  all  the  terms  after  the 
second,  as  being  very  small,  we  obtain  for  the  beginning  and  end  of  the  eclipse 
the  more  convenient  formula 

_  1800s.  (Aj_—  i//2) 
'-          "    B.    ' 


CALCULATION   OF  AN   ECLIPSE  OF  THE   SUN.  431 

If  the  moon's  apparent  latitude  be  found  for  the  approximate  time  of  greatest 
obscuration,  and  substituted  for  A'  in  equation  (Z),  this  equation  will  give  very 
nearly  the  apparent  distance  (a)  of  the  two  bodies  in  longitude  at  the  true  time 
of  greatest  obscuration.  With  this,  and  the  apparent  distance  at  the  approximate 
time  of  greatest  obscuration,  together  with  the  relative  apparent  motion  in  longi- 
tude, the  true  time  of  greatest  obscuration  may  be  found  nearly  by  simple  propor- 
tion. A  more  accurate  result  may  then  be  had  by  finding  the  moon's  apparent 
latitude  for  the  time  obtained,  substituting  it  for  A.'  in  equation  (I)  and  then  repeat- 
ing the  calculations, 

A  simpler,  though  less  accurate  method  than  that  already  given,  of  finding  the 
times  of  beginning  and  end  of  the  total  or  annular  eclipse,  is  to  compute  the  half 
duration  of  the  total  or  annular  eclipse,  and  add  it  to,  and  subtract  it  from,  the 
time  of  greatest  obscuration.  This  interval  may  easily  be  determined,  if  we  can 
find  the  rate  of  motion  on  the  relative  orbit,  and  the  distance  passed  over  by  the 
moon's  centre  during  the  interval.  Let  g,  g  (Fig.  134)  be  the  places  of  the  moon!s 
centre  at  the  instants  of  the  two  interior  contacts,  and  MT&,  the  distance  passed 
over  hi  some  short  interval  (L).  Let  0  —  <  Mw&  the  complement  of  the  inclination 
of  the  relative  orbit,  k  =  MA-.  k'  =  Mw,  and  t  =  half  duration  of  total  or  annular 
eclipse.  The  triangles  MwA-,  CRM,  give 

'  =  .-L.  .   .(m): 


sin  M«&  sin 

T?\f  V 

and  tang  RCM  =  tang  Mnfc  =  —  ,  or,  tang  0  =  _  .  .  .  (TO). 

CR  o- 

Finding  the  value  of  fl  by  the  last  equation,  and  substituting  it  in  equation  (m 
we  obtain  the  value  of  k'  ;   and  then,  to  find  t,  we  have 

k':L::  ty  :  t,  or  t  =  ±JL*L 


whence,  t  =  -        =  -  A)  .  .  .  (o) 

/c  k 

The  apparent  distance  of  the  centres  of  the  two  bodies  at  the  time  of  greatest  ob- 
scuration being  known,  the  quantity  of  the  eclipse  may  be  readily  found.  We  have 
but  to  subtract  the  apparent  distance  from  the  sum  of  the  apparent  semi-diameters, 
and  state  the  proportion,  as  the  sun's  apparent  diameter  :  the  remainder  ::  12  digits  : 
the  digits  eclipsed.  (For  a  more  particular  description  of  the  method  of  calculat- 
ing a  solar  eclipse,  see  Prob.  XXX.) 

CALCULATION  OF  AN  OCCULTATION. 

The  calculation  of  an  occultation  is  very  nearly  the  same  as  that  of  a  solar  eclipse. 
The  only  difference  is  in  the  data.  The  star  has  no  diameter,  parallax,  or  motion 
in  longitude  ;  and  as  it  is  situated  without  the  ecliptic,  we  have,  in  place  of  the  lati- 
tude of  the  moon,  employed  in  solar  eclipses,  the  difference  between  the  latitude  of 
the  moon  and  that  of  the  star,  and  in  place  of  the  difference  between  the  longitudes 
of  the  two  bodies  and  their  relative  hourly  motion  in  longi- 
tude, these  quantities  referred  to  an  arc  passing  through  the 
star  and  parallel  to  the  ecliptic.  Thus,  if  EC  (Fig.  133)  re- 
present the  ecliptic,  K  its  pole.  5  the  situation  of  the  star, 
M  that  of  the  moon,  and  sm  an  arc  passing  through  *  and 
parallel  to  the  arc  EC,  we  have  in  place  of  ??iM,  m'M  = 
m^  —  mm',  and  in  place  of  Sw,  sm.  The  hourly  variation 
of  Sm  must  also  be  reduced  to  the  arc  xm'. 

The  reduction  of  the  difference  of  longitude  of  the  moon 
and  star,   to   the  parallel  to   the  ecliptic,  passing  through 
the  star,  is  effected  by  multiplying  the  difference  by  the  co- 
sine of  the  latitude  of  the  star.    For,  let  AB  (Fig.  135)  be 
135.  an  arc  of  the  ecliptic,  and  A'B'  the  corresponding  arc  of  a 


432  APPENDIX. 

circle  paiallel  to  it,  then,  since  sirm'lar  arcs  of  circles  are  proportional  to  their 
radii  we  have 

BC  :  B'C'  ::  AB  :  A'B'=  AB-B'C' 
BG 

But,  B'C'  =  Ca  =  B'C  cos  BCB'  =  BC  cos  BB': 

hence,  A'B"  =  A*B°Bg»  BB'  =  AB  cosBB". 

The  reduction  of  the  relative  hourly  motion  in  longitude  to  the  parallel  in  question, 
is  obviously  effected  in  the  same  manner. 


NOTE  I. 

CONSTRUCTION  OF  TABLES. 

The  determination  of  the  place  of  the  sun  or  moon,  or  of  a  planet,  may  be  greatly 
facilitated  by  the  use  of  tables.  The  principle  and  mode  of  construction  of  tables 
adapted  to  this  purpose  are  nearly  the  same  for  each  body. 

We  will  first  explain  the  mode  of  constructing  tables  for  facilitating  the  computa- 
tion of  the  sun's  longitude.  We  have  the  equation 

True  long.  =  mean  long.  +  equa.  of  centre  +  inequalities  +  nutation. 

If,  then,  tables  can  be  constructed  which  will  furnish  by  inspection  the  mean  longi- 
tude, the  equation  of  the  centre,  the  amounts  of  the  various  inequalities  in  longitude, 
and  the  nutation  in  longitude,  at  any  assumed  time,  we  may  easily  find  the  true 
longitude  at  the  same  time. 

(1.)  For  the  mean  longitude. — The  sun's  mean  motion  in  longitude  'in  a  mean  tropi- 
cal year,  is  360°.  From  this  we  may  find  by  proportion  the  mean  motions  in  a  com- 
mon year  of  365  days,  and  a  bissextile  year  of  366  days. 

With  these  results,  and  the  mean  longitude  for  the  epoch  of  Jan.  1,  1850  (see 
Table  II.),  we  may  easily  derive  the  mean  longitude  at  the  beginning  of  each  of  the 
years  prior  and  subsequent  to  the  year  1850.  The  second  column  of  Table  XVIII. 
contains  the  mean  longitude  of  the  sun  at  the  beginning  of  each  of  the  years  in- 
serted in  the  lirst  column.  The  third  column  of  this  table  contains  the  mean  longi- 
tude of  the  perigee  at  the  same  epochs :  it  was  constructed  by  means  of  the  mean 
longitude  of  the  perigee  found  for  the  beginning  of  the  year  1800,  and  its  mean 
yearly  motion  in  longitude.* 

Having  the  sun's  mean  daily  motion  in  longitude,  we  obtain  by  proportion  the 
motion  in  any  proposed  number  of  months,  days,  hours,  minutes,  or  seconds.  Table 
XIX  contains  the  respective  amounts  of  the  sun's  motion  from  the  commencement 
of  the  year  to  the  beginning  of  each  month ;  Table  XX,  the  sun's  mean  motion 
from  the  beginning  of  any  month  to  the  beginning  of  any  day  of  the  month,  and 
his  motion  for  hours  from  1  to  24 ;  and  Table  XXI,  the  same  for  minutes  and 
seconds  from  1  to  60.  With  these  tables  the  sun's  mean  motion  in  longitude  in  the 
interval  between  any  given  time  iu  any  year  and  the  beginning  of  the  year  may 
be  had:  and  if  this  be  added  to  the  epoch  for  the  given  year,  taken  out  from  Table 
XVIII,  the  result  will  be  the  mean  longitude  at  the  given  time.  (See  Problem 
IX.) 

Tables  XIX  and  XX  also  contain  the  motions  of  the  sun's  perigee,  from  which 
and  the  epoch  given  by  Table  XVIII  results  the  longitude  of  the  perigee  at  any 
proposed  time.  The  longitude  of  the  perigee  is  given  in  the  Solar  Tables  for  the 
purpose  of  making  known  the  mean  anomaly,  the  mean  anomaly  being  equal  to  the 
mean  longitude  minus  the  longitude  of  the  perigee. 

(2.)  For  the  equation  of  the  centre. — To  find  the  equation  of  the  centre  of  an  orbit 
we  have  the  following  equation : 

Equa.  of  centre  —  A  sin  9  +  B  sin  29  +  C  sin  30  +  &c. ; 

*  The  quantities  in  Table  XVIII  are  called  Epochs.  The  Epoch  of  a  quantity 
is  its  value  at  some  chosen  epoch. 


CONSTRUCTION  OF  TABLES.  433 

in  which  A,  B.  C,  Ac.,  are  constants  that  rapidly  decrease  in  value,  and  which  may 
be  determined  for  any  particular  orbit,  and  0  the  mean  anomaly.  Now,  by  giving  to 
the  mean  anomaly  9  in  this  equation  a  series  of  values  Increasing  by  small  equal 
differences  (of  1°,  for  instance,)  from  zero  to  36o°,  ai;d  computing  the  correspond- 
ing values  of  the  equation  of  the  centre,  then  registering  in  a  column  the  different 
values  assigned  to  0,  and  in  another  column  to  the  ruht  of  this  the  computed  values 
of  the  equation  of  the  centre,  we  shall  obtain  a  table  which  will  give  on  inspection 
the  equation  of  the  centre  corresponding  to  any  particular  mean  anomaly.  In  this 
manner  was  constructed  Table  XXV.  In  '.his  table,  however,  for  the  sake  of  com- 
pactness, the  values  of  the  equation,  instead  of  being  registered  in  one  column,  are 
put  in  as  many  different  columns  as  there  may  be  different  numbers  of  signs  in  tho 
value  of  the  mean  anomaly ;  each  column  answering  to  the  particular  number  of 
signs  placed  at  the  head  of  it. 

If  the  equation  of  the  centre  at  an  assumed  time  be  required,  find  the  mea.n 
anomaly  by  the  tables,  and  with  the  value  fuund  for  it  take  out  the  equation  of  the 
centre  from  Table  XXV. 

The  given  quantity  with  which  a  quantity  is  taken  from  a  table  is  called  the 
Argument  of  that  quantity.  Accordingly  the  mean  anomaly  is  the  argument  of 
the  equation  of  the  centre  in  Table  XXV. 

(3.)  For  the  inequalities. — The  equations  of  the  inequalities,  as  we  have  already 
stated,  are  of  the  form  C  sin  A,  the  argument  A  being  the  difference  between  the 
longitude  of  the  disturbing  planet  and  that  of  the  earth,  or  some  multiple  of  this 
difference.  With  the  equations  of  the  inequalities  a  table  of  eacli  inequality  may 
be  constructed,  upon  the  same  principles  as  Table  XXV.  But,  as  the  expression 
for  the  whole  perturbation  in  longitude  (Art.  212),  produced  by  any  one  planet, 
involves  only  two  variables,  the  longitude  of  the  earth  and  the  longitude  of  the  planet, 
it  is  thought  to  be  more  convenient  to  have  a  table  of  double  entry,  which  will  give 
the  amount  of  the  perturbation  by  means  of  the  two  variables  as  arguments. 
Such  a  table  may  be  constructed,  by  assigning  to  the  longitude  of  the  earth  and 
the  longitude  of  the  disturbing  planet  a  series  of  values  increasing  by  a  common 
difference,  and  computing  with  each  set  of  the  values  of  these  quantities,  the  cor- 
responding amount  of  the  perturbation. 

In  connection  with  the  tables  of  the  perturbations,  we  must  have  tables  that 
make  known  the  values  of  the  arguments  at  any  given  tune.  Now,  the  mean  lon- 
gitude of  the  sun  may  be  found  by  the  solar  tables  (XVIII  to  XXI),  and  theuoe 
the  mean  heliocentric  longitude  of  the  earth  by  subtracting  ISO3 ;  and  the  mean 
longitude  of  the  disturbing  planet  may  be  had  from  similar  tables.  The  columns 
of  Table  XVIII,  marked  I,  II,  III,  IV,  V,  VI,  VII,  contain  the  arguments  of  all 
the  perturbations  for  the  beginning  of  each  of  the  years  registered  in  the  first 
column,  expressed  in  thousandth  parts  of  a  circle.  Tables  XIX  and  XX  con- 
tain the  variations  of  the  arguments  for  months,  days,  and  hours.  Their  varia- 
tions for  minutes  and  seconds  are  too  small  to  be  taken  into  account.  With  these 
tables,  and  Table  XV HI.  the  values  of  the  arguments  at  any  given  time  may  bo 
found,  and  by  means  of  the  arguments  the  perturbations  may  be  taken  from 
Tables,  XXVIII,  XXIX,  XXXII,'  XXXI,  XXX,  and  XXXIII. 

(4.)  For  the  nutation. — The  formula  for  the  lunar  nutation  in  longitude,  is  17".E 
sin  N— 0  '.2  sin  2  N,  in  which  N  denotes  the  supplement  (to  360°)  of  the  longitude- 
of  the  moon's  ascending  node.  With  this  formula  the  second  column  of  the  Tabte 
XXVII  was  constructed.  The  value  of  N,  in  thousandth  parts  of  a  circle,  result1* 
from  Tables  XVIII,  XIX.  and  XX.  The  solar  nutation  is  also  givcu  by  Table 
XXVII. 

Tables  may  also  be  constructed  that  will  facilitate  the  computation  of  the  radius- 
vector.  We  have 

True  rad.-vector  =  elliptic  rad. -vector  +  perturbations. 

A  table  of  the  elliptic  radius  vector  may  be  formed  by  means  of  the  polar 
equation  of  the  orbit,  and  tables  of  the  perturbations  from  their  analytical  expres- 
sions (Art.  214).  The  tables  ot"  the  perturbations  will  have  the  same  arguments 
as  the  tables  of  the  perturbations  of  longitude. 

Lunar  and  planetary  tables  are  constructed  upon  the  same  principles  as  the  solar 
'.ables  we  have  been  describing,  which  serve  to  make  known  the  orbit  longitude 
and  radius- vector.  But  other  tables  are  necessary  in  the  case  of  these  bodies,,  for 
the  computation  of  the  ecliptic  longitude  and  the  latitude. 

28 


434  APPENDIX. 

The  difference  between  the  orbit  longitude  and  the  ecliptic  longitude  is  called  the 
Reduction  to  the  ecliptic.  A  formula  for  the  reduction  lias  been  investigated, 
in  wliich  the  variable  is  the  difference  between  the  orbit  longitude  and  the  lon- 
gitude of  the  node  (or,  what  amounts  to  the  same,  the  orbit  longitude  plus  the 
supplement  of  the  longitude  of  the  node  to  300°).  If  this  formula  be  reduced  to 
a  table,  by  taking  the  reduction  from  the  table  and  adding  it  to  the  orbit  longitude, 
we  shah1  have  the  ecliptic  longitude.  Table  LIII  is  a  table  of  reduction  for  the 
moon. 

For  the  latitude,  we  have  the  equation 

True  lat.  =  lat.  in  orbit  +  perturbations. 

We  have  already  seen  (Art.  20 1)  that 

sin  (lat.  in  orbit)  =  sin  (orbit  long.  — long,  of  node)  sin  (iuch'na.) 

A  table  constructed  from  this  formula  will  have  for  its  argument  the  orbit  lon- 
gitude minus  the  longitude  of  the  node,  which  is  also  the  argument  of  the  reduction. 
(See  Table  LV.) 

The  tables  of  the  perturbations  in  latitude  are  constructed  upon  the  same  prin- 
ciples as  the  tables  of  the  perturbations  in  longitude  and  radius-vector. 


NOTE  II. 

(Referred  to  on  p.  175.) 

The  fact  stated  in  the  text  (Art.  273)  that  the  penumbra  of  the  solar  spot  doeg 
not  begin  to  be  formed  until  the  spot,  which  at  first  has  an  umbral  blackness,  has 
attained  a  measurable  size,  is  cited  by  astronomical  writers  as  a  circumstance 
strongly  favoring  the  hypothesis  of  the  origination  of  the  spot  in  a  disturbance  from 
below.  But  this  fact  may  be  reconciled  with  the  opposite  hypothesis  advocated  in 
the  text,  if  we  reflect  that  the  penumbral  lies  at  a  considerable  depth  below  the 
luminous  envelope,  and  that  the  process  of  discharge  and  ascent  of  a  column  of 
photospheric  matter  (Art.  293),  which  results  in  disclosing  to  view  a  portion  of  the 
penumbral  envelope,  should  occupy  a,  certain  interval  of  time  in  passing  down  to  it. 
During  this  interval  this  envelope  may  have  an  umbral  blackness,  and  it  may  owe 
its  subsequent  visibility,  as  distinct  from  the  umbra,  entirely  to  the  fact  that  it 
acquires  a  luminosity  in  consequence  of  the  electric  discharges  that  attend  the 
process  of  spot  evolution,  which  has  penetrated  to  its  depth  in  the  atmosphere  of 
the  sun.  This  view  is  supported  by  the  fact  that  it  furnishes  a  simple  explanation 
of  the  decrease  in  the  brightness  of  the  penumbra  from  the  umbra  to  its  outer 
margin.  We  have  only  to  observe  that  the  process  of  expulsion  and  ascent  of 
vaporous  matter,  which  begins  at  a  certain  point  of  the  photosphere,  at  the  same 
time  that  it  proceeds  downward,  spreads  laterally,  and  that  when  it  has  penetrated 
to  the  depth  of  the  penumbral  envelope  at  the  point  below  that  where  it  originated, 
an  opening  of  a  certain  size  will  have  been  formed  in  the  luminous  envelope,  and 
except  below  the  central  portions  of  this  opening,  the  lower  envelope  will  still  be 
in  a  comparatively  quiescent  condition,  and  retain  its  natural  depth  of  shade. 
Subsequently  the  same  process  of  evolution  is  repeated  at  this  envelope ;  an  open- 
ing is  made  in  it  that  has  the  umbral  depth  of  shade,  and  this  is  surrounded  by  a 
region  of  luminous  activity,  which  is  the  penumbra  of  the  spot,  and  is  fringed  by 
a  dark  border  consisting  of  the  part  which  the  descending  action  has  not  yet 
readied.  In  the  case  of  the  larger  spots  and  of  long  continuance,  the  same  pro- 
cess penetrates  to  the  third  envelope,  and  the  former  umbra  shows,  in  its  turn,  a 
black  central  nucleus,  surrounded  by  a  fringe  of  a  shade  perceptibly  less  dark. 

Upon  the  present  hypothesis  with  regard  to  the  mode  of  development  of  the 
spots,  the  principal  varieties,  consisting  of  a  spot  without  a  penumbra,  a  spot  without 
an  umbra,  a  spot  without  the  central  black  nucleus  at  the  centre  of  the  umbra,  and  a 
spot  with  this  nucleus,  are  but  the  varieties  that  present  themselves  according  as  the 
process  of  discharge,  beginning  at  the  surface  of  the  photosphere,  penetrates  only 
through  the  first  envelope,  or  through  the  first  and  as  far  as  the  second,  or  through 


DEVELOPMENT  OF  SUN'S  SPOTS.  435 

the  first  and  second,  or  through  the  first,  second,  and  third.  Upon  the  old  hypothesis, 
it  is  necessary,  in  order  to  explain  these  diverse  phenomena,  to  assume  that  there 
are  three  possible  centres  of  explosive  action,  posited  below  the  successive 
envelopes. 

So  long  as  the  active  evolution  continues  at  the  lower  envelopes,  the  ascending 
vaporous  column,  expanding  as  it  rises,  should  check  any  eventual  tendency  of  the 
opening  in  the  luminous  envelope  to  close.  "When  the  activity  subsides  at  the 
penumbra!  stratum,  and  the  opening  in  it  begins  to  close,  this  should  be  followed 
by  a  similar  collapse  in  the  regions  above  it ;  and  so  the  contraction  of  a  spot 
should  generally  begin  in  accordance  with  observation  at  the  umbra,  and  be  fol- 
lowed by  the  encroachment  of  the  luminous  margin  upon  the  penumbra.  But  it 
is  conceivable,  also,  that  the  closing  up  of  a  spot  may  result  from  a  condensation 
into  luminous  clouds  of  portions  of  the  expanded  matter  ascending  within  the 
region  of  the  spot;  and  that  the  luminous  veil  that  is  often  seen  to  form  over  a 
spot,  and  the  bridges  of  light  which  suddenly  span  the  umbra,  are  the  first  indica- 
tions of  the  beginning  of  such  condensation.'-:. 

To  give  a  more  complete  exposition  of  the  author's  theory  of  the  development 
of  solar  spots,  the  following  general  conclusions  are  added  to  those  given  in 
the  text. 

1.  The  spots  are,  for  the  most  part,  due  to  diminutions  occurring  in  the  magnetic 
intensity  that  obtains  in  the  photosphere  of  the  sun. 

2.  Each  planet  operates  on  the  photosphere  by  repulsive  impulses,  and  develops 
electro-magnetic  currents  circulating  in  a  direction  opposite  to  that  of  the  rotation. 
The  effective  currents  thus  originating  are  differential,  and  result  from  the  fact, 
that  on  the  left  or  east  side  of  the  line  from  the  planet  to  the  sun's  centre  the 
motion  of  the  surface  is  in  a  direction  opposite  to  that  in  which  the  impulses  are 
propagated,  and  on  the  other  side  in  the  same  direction. 

3.  The  general  tendency  of  such  new  currents  should  be  to  increase  the  mag- 
netic intensity  at  the  surface  of  the  photosphere  ;  but  it  is  possible  that  in  peculiar 
conditions  of  the  photospheric  matter,  as  to  density  and  other  qualities,  the  super- 
ficial currents  developed  by  planetary  action  may  prevail  over  those  set  in  motion 
below  the  surface,  and  the  opposite  magnetic  effect  be  produced. 

4.  The  motion  of  the  sun  through  space  alt»o  develops  similar  magnetic  cur- 
rents, which  should  have  a  similar  effect.     These  currents  may  be  considered  as 
originating  on  that  side  of  the  sun  where  the  absolute  motion  of  a  point  of  its 
surface  is  the  most  rapid. 

5.  If  the  sun  were  stationary  the  motion  of  revolution  of  a  single  planet  would 
have  but  little  direct  effect  to  change  its  magnetic  action  on  the  sun  as  a  whole ; 
except  so  far  as  this  may  vary  by  reason  of  the  varying  distance  of  the  planet.     But 
in  point  of  fact  the  effective  action  of  a  planet  will  change  with  its  distance  in  longi- 
tude from  the  point  towards  which  the  sun  is  moving  in  space.     It  will  be  at  its 
maximum  when  the  planet  is  in  heliocentric  conjunction  with  this  point,  and  at  a 
minimum  when  it  is  in  opposition  to  it.    In  the  former  case  its  heliocentric  longitude 
will  be  253°,  and  in  the  latter  73°. 

6.  The  joint  magnetic  action  of  two  planets  varies  with  their  relative  position ; 
it  has  its  maximum  value  when  the  planets  are  in  conjunction,  and  its  minimum 
when  they  are  in  opposition. 

7.  The  epochs  of  the  conjunction  of  one  planet  with  another,  or  of  a  planet  with 
the  point  towards  which  the  sun  is  moving,  are,  in  general,  the  epochs  of  minimum 
of  spots,  since  the  magnetic  intensity  at  the  surface  of  the  photosphere  is  on  the  increase 
before  every  such  epoch.     The  approximate  coincidence  of  planetary  conjunctions 
with  special  epochs  of  minimum  of  spots  is  a  recognized  lact  in  the  case  of  Jupiter 
and  the  earth,  and  Venus  and  the  earth.     On  the  other  hand,  the  opposition  of  two 
planets  tends  to  determine  a  maximum  of  spots. 

8.  Jupiter  and  Venus  are  the  most  influential  planets.    The  period  of  the  spots  is  de- 
termined mainly  by  the  motion  of  revolution  of  Jupiter,  but  appears  to  be  modified  by 
the  varying  planetary  configurations,  and  also  by  changes  occurring  in  the  physical 
condition  of  the  sun's  photosphere.     The  varying  action  of  Jupiter,  in  the  course  of 
a  revolution,  has  been  attributed  to  its  variations  of  distance  from  the  sun,  but  it 
seems  improbable  that  effects  so  marked  should  result  from  so  slight  a  cause.      The 
mean  period  of  the  spots  is,  according  to  Wolf,  nearly  one  year,  and  according  to 
Schwabe.  nearly  two  years  less  than  Jupiter's  period  of  revolution.     This  difference 
cannot  be  explained  by  any  mere  recurrence  of  planetary  configurations.     In  fact, 


436 


APPENDIX. 


epochs  of  maximum  and  minimum  of  spots  have  occurred  under  every  variety  of 
configurations ;  and  also  when  Jupiter  has  had  every  variety  of  position  in  its  orbit. 
The  following  Table  shows  the  mean  positions  of  Jupiter  and  Saturn  at  various  such 
epochs,  from  17  50  to  1860;  together  with  the  relative  numbers  showing,  according 
to  Wolf,  the  spot-condition  of  the  sun. 


Mean  Heliocentric 

Mean  Heliocentric 

Longitude. 

Longitude. 

Epochs  of 
Maxima. 

Relative 
Numbers. 

Jupiter. 

Saturn. 

Epochs  of 
Minima. 

Obs'd  Min. 
-Mean  Min. 

Jupiter. 

Saturn. 

1750.0 

68.2 

4° 

231° 

1744.5 

+  0.558 

197° 

164° 

17.61.5 

75.0 

353 

12 

1755.7 

+  0.639 

177 

301 

1770.0 

79.4 

251 

116 

1766.5 

+  0.^20 

145 

73 

1779.5 

99.2 

179 

232 

1775.8 

—1.499 

67 

187 

1788.5 

90.6 

93 

342 

1784.8 

—3.618 

340 

297 

1804.0 

70.0 

203 

172 

1798.5 

—1.037 

36 

105 

1816.8 

45.5 

232 

329 

1810.5 

-0.156 

41 

251 

1829.5 

53.5 

258 

124 

1823.2 

+  1.425 

66 

47 

1837.2 

111.0 

131 

218 

1833.8 

+  0.906 

28 

177 

1848.6 

100.4 

117 

358 

1844.0 

—0.013 

338 

301 

1860.5 

98.6 

119 

143 

1856.2 

+  1.068 

348 

91 

It  \vi!l  be  seen  that  since  1761  the  maxima  have  occurred  when  Jupiter  was  in 
the  secoi id  or  third  quadrant  of  longitude;  and  that  since  1755,  the  minima  have 
occurred  when  he  was  in  the  other  two  quadrants.  But  previous  to  those  dates  the 
condition  of  things  was  reversed,  and  the  transition  from  the  one  condition  to  the 
other  was  gradual.  This  fact  seems  to  necessitate  the  supposition  that  the  physical 
condition  of  the  sun's  photosphere  is  liable  to  changes,  by  reason  of  which  the  ordi- 
nary effect  of  the  planets  is,  at  intervals,  wholly  reversed. 

9.  The  highest  maxima  of  spots  have  occurred  when  Jupiter  was  in  the  second 
quadrant  of  longitude;  that  is  after  he  has  passed  a  certain  distance  beyond  the 
point  (long.  73°)  where  his  magnetic  action  on  the  sun  is  most  directly  opposed  to 
the  effect  of  the  sun's  motion  through  space,  and  advanced  towards  the  aphelion  of 
the  orbit,  where  the  direct  magnetizing  tendency  of  the  planet  is  the  least. 

It  should  here  be  remarked  that  the  author  has  undertaken,  in  other  publications,  to 
show  that  the  earth  derives  its  magnetic  condition  from  the  collision  of  the  molecules 
of  its  revolving  and  rotating  mass  with  the  ether  of  space,  and  that  the  sun  and  the 
plau-ets  should  each  be  in  a  magnetized  condition  from  a  similar  cause ;  also  that  in 
the  new  terrestrial  magnetic  currents  being  continually  developed  by  the  earth's 
motions  of  revolution  and  rotation  combined,  in  connection  with  those  generated  in 
the  photosphere  (upper  atmosphere)  of  the  earth  by  ethereal  impulses,  and  streams 
of  nebulous  matter,  proceeding  from  the  sun,  we  have  the  principal  operative  causes 
of  the  disturbances  experienced  by  the  magnetic  needle  on  the  earth's  surface. 


NOTE  III. 

(Referred  to  on  page  275.) 

A  remarkable  analogy  in  the  periods  of  rotation  of  the  primary  planets  was  dis- 
covered a  few  years  since  (1848)  by  Daniel  Kirkwood,  of  Pottsville,  Pennsylvania. 
This  analogy  is  now  generally  known  by  the  name  of  Kirkwood's  Law,  and  is  as 
follows : 

"  Let  P  be  the  point  of  equal  attraction  between  any  planet  and  the  one  next  in- 
terior, the  two  being  in  conjunction :  P'  that  between  the  same  and  the  one  next 
exterior. 

Let  also  D  =  the  sum  of  the  distances  of  the  points  P,  P'  from  the  orbit  of  the 


planet;  which  I  shall  call  the  diameter  of  the  sphere  of  the  planet's  attraction; 
D'  =  the  diameter  of  any  other  planet's  sphere  of  attraction  found  in  like  man- 


ner; 


KIRKWOOD'S  ANALOGY. 


487 


n  =  the  number  of  sidereal  rotations  performed  by  the  former  during  one  sidereal 
revolution  around  the  sun  ; 

n'  =  the  number  performed  by  the  latter  ;  then  it  will  be  found  that 

n>  :  »'«  ::  D3.-  D' 


or  n  =  n'  jfr. 

That  is,  the  square  of  the  member  of  rotations  made  by  a  planet  during  one  revolution 
round  the  sun,  is  piopnrttonal  to  the  cube  of  the  diameter  of  its  sphere  of  attraction; 

or  —  ^-is  a  constant  quantity  for  all  the  planets  of  the  solar  system. 

The  analogy  thus  announced  has  been  subjected  to  a  rigid  mathematical  exami- 
nation by  Sears  C  Walker,  with  th«?  following  result  :  ""We  may  therefore  conclude," 
says  he,*  "that  whether  Kirkwood's  Analogy  is  or  is  not  the  expression  of  a  physi- 
cal law,  it  is  at  least  that  of  a  physical  fact  iu  the  mechanism  of  the  universe." 
(See  the  American  Journal  of  Science,  New  Series.  voL  x.  pp.  19-26.) 

There  are  but  three  planets,  viz.,  Venus,  the  Earth,  and  Saturn,  for  which  all  the 
elements  embraced  in  this  law  are  known.  The  diameters  of  the  spheres  of  attraction 
of  Mercury  and  Neptune  are,  from  the  nature  of  the  case,  incapable  of  determina- 
tion. The  mass  of  the  one  planet  into  which  the  planetods  are  supposed  once  to 
have  been  united  is  not  known  with  certainty,  as  there  maybe  planetoids  yet  undis- 
covered, and  its  period  of  rotation  is  hypothetical  only.  The  diameters  of  the  spheres 
of  attraction  of  Mars  and  Jupiter  can  only  be  approximately  determined  ;  and  the 
period  of  rotation  of  Uranus  is  unknown. 

The  interest  naturally  awakened  by  the  announcement  of  so  important  a  discov- 
ery was  heightened  by  the  fact,  that  it  was  at  once  perceived  that  it  furnished  a 
new  and  powerful  argument  in  support  of  the  nebular  hypothesis  (or  cosmogony) 
devised  by  Laplace.  (See  a  paper  on  this  subject  by  Dr.  B.  A.  Gould,  Jr.,  in  the 
Journal  of  Science,  New  Series,  vol.  x.  p.  26,  etc.) 


NOTE  IT. 


(Referred  to  on  page  2 7  7.) 

It  remains  to  deduce,  if  possible,  the  known  law  of  the  distribution  of  the  incli- 
nations of  the  cometary  orbits.  This  law  is,  that  the  number  of  orbital  inclinations 
of  different  values  increases  with  the  angle  from  0°  to  50°  or  60°,  and  then  de- 
creases ;  as  appears  from  the  following  tabular  statement,  from  which  the  comets 
of  short  period  have  been  deducted. 


0°  to  10° 

10°  to  20° 

20°  to  80°  80°  to  40° 

40°  to  50° 

50°  to  60° 

60°  to  70° 

70°  to  80° 

80«  to  900 

11 

15 

16 

28 

34 

32 

23 

22 

14 

Let  us  first  consider  the  case  of  a  discharge  from  the  equator,  and  conceive,  for 
the  present,  the  direction  of  discharge  to  be  tangent  to  the  surface.  Since  the 
orbits  of  the  class  of  comets  under  consideration  are  very  eccentric,  the  initial  ve- 
locity must  be  very  much  less  than  the  velocity  of  rotation  at  the  equator.  If  we 
fix  upon  a  maximum  limit  (V)  to  this  velocity,  at  any  assumed  epoch,  and  from 
this  and  the  velocity  of  rotation  at  the  equator  (v)  deduce  the  direction  of  the  ex- 
pelling force,  and  the  velocity  (t>°,)  due  to  its  action,  it  will  be  seen : 

(1.)  That  this  force  will  take  effect  in  directions  opposed  to  that  of  the  rotation, 
and  inclined  to  it  under  angles  differing  but  little  from  180°,  whatever  direction 
may  be  assumed  for  V,  the  resultant  or  effective  velocity. 

(•2.)  That  the  velocity,  v',  due  to  the  expelling  force,  will  be  either  equal  to  the 
velocity  of  rotation,  v,  or  a  little  greater,  or  a  little  less. 


438  APPENDIX. 

Under  these  circumstances,  nebulous  masses  may  be  projected  in  every  variety 
of  direction  in  the  tangential  plane,  which  would  move  in  planes  having  any  angle  of 
inclination,  and  describe  orbits  of  every  variety  of  eccentricity  greater  than  that 
answering  to  the  assumed  maximum  initial  velocity,  V. 

Now,  if  we  conceive  the  point  at  which  a  discharge  occurs  to  lie  in  any  latitude 
(l\  the  velocity  of  rotation  will  be  less  in  the  proportion  of  cos  I  to  1  ;  and  making 
use  of  the  parallelogram  of  velocities  as  before,  and  retaining  the  same  effective  ve- 
locity Y,  we  find  that  for  any  assumed  direction  of  V,  the  line  of  direction  of  the 
expelling  force  will  deviate  more  from  that  of  direct  opposition  to  the  motion  of 
rotation  than  at  the  equator ;  and  that  the  deviation  will  increase  with  the  latitude. 
For  any  latitude  (I)  the  orbits  described  by  the  masses  detached  may  have  every 
variety  of  inclination  from  1°  to  90° ;  but  the  larger  inclinations  will  result  from  an 
action  of  the  expelling  force  exerted  in  a  direction  inclined  under  a  smaller  angle  tc 
the  meridian  in  proportion  as  I  is  greater. 

If  we  conceive  this  force  to  act  in  some  direction  oblique  to  the  surface,  instead 
of  tangentially,  the  velocity  due  to  its  action  will  be  replaced  by  its  tangential 
component,  which  is  now  to  be  taken  equal  to  V.  The  aphelion  of  the  orbit  will 
also  now  be  removed  to  a  certain  distance  from  the  nebulous  body,  instead  of  be- 
ing within  its  surface  at  the  point  of  discharge. 

In  view  of  what  has  now  been  stated,  it  may  be  seen  that  the  actual  law  of  disiri- 
bution  of  the  inclinations  might  result  if  the  frequency  of  discharge  were  to  decrease 
with  the  angle  included  between  the  line  of  direction  of  the  operating  force  and  th& 
meridian. 

This  theoretical  result  suggests,  as  the  possible  origin  of  the  separation  of  frag- 
ments from  the  surface  of  the  nebulous  body,  the  flow  of  electric  currents  in  alt  di- 
rections from  points  near  the  equator  ;  similar  to  the  currents  we  have  conceived  to 
be  developed  by  planetary  action  on  the  sun's  photospheric  surface,  and  to  give 
rise  to  the  solar  spots  (293).  Such  currents,  in  proportion  to  the  resistance  they 
experience,  would  develop  statical  electricity,  the  repulsive  action  of  which  might 
occasion  discharges  in  the  direction  of  the  radial  currents  and  obliquely  upwards. 
Upon  this  conception,  if  we  consider  the  disengagements  that  may  occur  from  any 
point  of  any  one  meridian,  and  bear  in  mind  that  the  electric  currents  supposed  may 
proceed,  indifferently,  from  all  points  near  the  equator,  it  will  be  observed  that  the 
frequency  of  the  detachment  of  fragments  from  the  point  in  question  will  decrease 
in  proportion  as  the  radial  current,  or  direction  of  the  expelling  force,  makes  a  less 
angle  with  the  meridian.  For  an  arc  of  the  equator  (say  10°)  will  subtend,  at  the 
point  considered,  a  greater  angle  in  proportion  as  it  is  nearer  to  the  meridian.  At 
each  point  of  the  meridian,  therefore,  the  liability  to  expulsive  action  should  be 
greatest  in  directions  nearly  perpendicular  to  the  meridian ;  for  whicb  directions 
the  resulting  orbital  inclinations  would  be  nearly  equal  to  the  latitude  of  the  point. 
The  effective  directions  of  expulsion  would  fall,  for  each  latitude,  /,  on  the 
meridian,  between  1°  and  90°.  The  complete  result  for  all  points  on  the  meridian, 
should  then  be  that  the  number  of  inclinations  would  augment  with  the  angle  of 
inclination  up  to  a  certain  large  value  of  this  angle,  and  then  decrease ;  in  accor- 
dance with  the  observed  law.  The  agreement  would  apparently  be  more  exact  if, 
as  would  naturally  be  supposed,  the  frequency  of  discharge  became  less  at  the 
higher  latitudes. 

We  must  suppose  that  the  masses  discharged  which  have  since  become  perma- 
nent members  of  the  solar  system,  must,  after  berng  projected  into  space,  have 
become  condensed  sufficiently  before  returning  within  the  attenuated  mass  of  the 
nebulous  body,  to  have  pursued  their  course  unaffected  by  the  resistance  of  the 
medium  traversed,  except  within  certain  limits. 

It  appears  that  whether  we  consider  the  movements  of  the  magnetic  needle  upon 
the  earth,  resulting  from  solar  action,  or  the  development  of  spots  on  the  sun's  sur- 
face by  planetary  influence,  or  the  rise  of  nebulous  envelopes  from  the  nucleus  of 
a  large  comet,  under  the  operation  of  the  sun,  or  the  origination  of  cometary  bodies, 
we  are  conducted,  in  each  instance,  to  the  primary  conception  of  electric  currents 
radiating  from  points  near  the  equator  of  the  body  subject  to  external  influence,  as 
playing  an  important  part  in  the  production  of  the  phenomena  observed.  Analogy 
would  then  lead  us  to  infer  that  the  excising  cause  of  the  electric  currents  sup- 
posed to  have  furnished  the  operating  cause  of  the  detachment  of  cometary  masses 
from  the  same  nebulous  body  from  which  the  planets  have  been  derived,  has  been 
an  action  of  the  planets  upon  the  surface  of  this  body,  similar  to  that  which  has 


ORIGIN   OF  SIDEREAL  SYSTEMS.  439 

been  operative  in  the  other  cases.  The  same  process  may  have  continued  in  opera- 
tion down  to  the  present  epoch,  originating,  in  the  later  ages,  meteorites  rather 
than  true  cometary  bodies.  In  fact  we  have  seen  that  a  continual  process  of  dis- 
charge of  nebulous  magnetic  matter  from  the  sun  is  in  operation  in  the  present 
age.  (Art  293.) 


NOTE  Y. 
ORIGIN  OF  SIDEREAL  SYSTEMS. 

"We  propose,  in  the  present  note,  to  develop  very  briefly  a  theory  of  the  possi- 
ble evolution  of  all  sidereal  systems  from  primordial,  rotating,  nebulous  masses. 

Fundamental  hypothesis,  and  circumstances  of  evolution.  Let  us  assume  that  the 
component  stars  of  every  cluster  were  originally  integrant  portions  of  a  vast  nebu- 
lous body,  and  that  this  body  had  a  motion  of  rotation  about  an  axis.  It  is  obvi- 
ous that  every  portion  of  the  rotating  mass  that  might  become  detached  would 
thenceforward  tend  to  revolve  independently  about  its  centre  in  a  certain  orbit 
Every  such  orbit  would  cross  the  plane  of  the  equator  in  two  points,  or  nodes , 
unless  the  detachment  occurred  in  that  plane,  in  \\-hich  event  the  orbit  would  lie 
within  it.  Again,  if  we  conceive  the  separation  to  have  taken  place  without  bring- 
ing disturbing  forces  into  play,  the  detached  portion  of  the  mass  should  have  an 
initial  motion  in  a  direQtion  parallel  to  the  equatorial  plane,  and  an  initial  velocity 
proportional  to  the  cosine  of  the  angular  distance  from  that  plane.  If  we  suppose 
impulsive  or  repulsive  forces  to  have  been  in  energetic  operation,  we  may  approxi- 
mately determine  the  nature  and  amount  of  their  influence  upon  these  initial  cir- 
cumstances, and  thence  upon  the  orbit  subsequently  described 

Case  I.  A  simultaneous  disruption  of  the  whole  mass  of  the  nebulous  body.  "We  will 
regard  the  original  nebulous  body  as  sensibly  spherical  in  form,  and  first  conceive  the 
disruption  to  have  occurred  at  the  same  epoch  in  all  its  parts ;  or  at  epochs  sepa- 
rated by  small  intervals  in  comparison  with  the  vast  duration  of  one  rotation 
period. 

General  results.  The  first  result  will  be  the  formation  of  a  globular  cluster  of 
stars,  separated  either  by  equal  or  unequal  intervals  of  space.  We  will  confine  our 
attention  to  the  case  of  equal  intervals.  All  the  stars  of  each  spherical  layer  will 
then  set  out  on  their  various  courses  at  the  same  epoch.  If  we  consider  those 
which  lie  on  any  one  meridian  of  the  outer  layer,  their  initial  velocities  will  de- 
crease proportionally  to  their  angular  distance  from  the  equator,  and  they  will 
therefore  set  out  in  elliptic  courses  that  will  be  more  eccentric  in  proportion  as  this 
distance  is  greater.  In  case  the  disruption  occurs  at  the  period  when  the  centri- 
fugal force  of  rotation  is  equal  to  the  force  of  gravity,  at  the  equator  of  the  nebu- 
lous body,  the  equatorial  stars  will  move  in  circles,  and  the  others  in  orbits  of  every 
degree  of  eccentricity,  from  a  circle  at  the  equator  to  a  right  line  at  the  poles. 
The  stars  in  question  will  all  pass,  at  the  end  of  one  quadrant  of  a  revolution, 
through  the  plane  of  the  equator  at  various  points  of  the  line  perpendicular  to  the 
plane  of  the  meridian  from  which  they  set  out.  Another  meridiau  of  the  outer  surface 
would  give  a  new  set  of  stars,  with  a  new  common  line  of  nodes  for  all  their  orbits. 
As  a  general  result  then  of  the  orbital  motions  communicated  to  the  stars  of  the 
outer  spherical  layer,  this  layer  witt  assume  the  form  of  an  ollate  spheroid,  with  its 
shorter  axis  coincident  with  the  line  of  the  original  axis  of  rotation.  If  we  consider 
the  next  spherical  layer  of  stars,  these  will  all  have  taken  up  their  independent 
movements  simultaneously  with  those  at  the  surface,  and  as  a  general  result  the 
whole^  layer  will  assume  the  oblate  spheroidal  form,  like  the  first.  The  same  will 
be  true  of  each  successive  layer,  and  the  contractions  will  proceed  simultaneously, 
while  the  order  of  the  layers  will  remain  unchanged. 

Periods  of  revolution.  Since  the  initial  velocities,  from  the  surface  inward,  are  pro- 
portional to  the  distance  from  the  axis  of  rotation,  and  the  attractive  forces  by 
which  the  stars  are  solicited  at  the  outset,  are  proportional  to  the  distance  from  the 
centre,  it  follows  that  all  stars  proceeding  from  points  at  the  same  angular  distance 
from  the  equator  will  accomplish  their  revolutions  in  the  same  period  of  tim&. 


440  APPENDIX. 

This  will  be  true  whether  the  accelerating  force  soliciting  each  star,  in  its  orbital 
motion,  varies  inversely  as  the  square  of  the  distance  from  the  centre,  or  directly 
as  the  distance ;  and  must  be  approximately  true  if  the  actual  law  of  variation  lies 
between  these  two.  Now  if  in  the  contraction  of  the  starry  layers  they  all  retained 
their  spherical  form,  each  star  would  be  subject  to  the  attraction  of  the  same 
spherical  mass,  during  an  entire  revolution,  and  therefore  the  law  of  variation  of 
its  accelerating  force  would  be  that  of  the  inverse  squares ;  but  in  point  of  fact 
the  layers  in  question  contract  into  spheroids  continually  increasing  in  oblateness, 
and  hence  the  accelerating  force  must  vary  according  to  a  less  rapid  law. 

Upon  the  supposition  that  the  law  of  the  inverse  squares  obtains,  the  period  of 
revolution  of  an  equatorial  star  in  a  circle,  would  be  to  that  of  a  star  from  very 
near  the  pole,  in  a  very  eccentric  ellipse,  as  2.8  to  1.  On  the  other  hand,  if  the 
force  varied  directly  as  the  distance,  the  two  periods  would  be  equal.  The  actual 
ratio  should  then  lie  between  2.8  and  1 ;  and  may  be  assumed  to  be  not  far  from 
2.  The  equatorial  star  would  complete  a  half  revolution  in  an  interval  of  time 
equal  to  the  duration  of  one  oscillation  of  an  ideal  celestial  pendulum,  having  a 
length  equal  to  the  radius  of  the  globular  cluster  at  the  epoch  of  its  formation, 
and  solicited  by  the  force  of  gravitation  in  operation  at  the  surface  of  the  cluster. 
Either  pole  of  the  cluster  would  contract  to  the  equatorial  plane,  and  attain  to  its 
limit  of  expansion  on  the  opposite  side,  while  such  an  ideal  pendulum  is  completing 
a  half  oscillation,  or  thereabouts.  Every  such  dynamical  cluster  has  its  vast  cycle, 
at  the  close  of  which  all  its  stars  return  approximately  to  their  original  positions. 
Such  a  state  of  things  would  be  realized  in  about  the  interval  of  two  periods  of 
revolution  of  the  equatorial  stars,  supposed  to  move  in  circles,  or  in  four  oscilla- 
tions of  the  representative  pendulum. 

Increase  of  density.  As  the  contraction  of  the  original  globular  cluster  pro- 
ceeds, the  density  continually  increases,  and  attains,  at  any  part  of  the  equato- 
rial plane,  its  greatest  value  at  the  epoch  when  the  stars  are  crossing  the  plane  in 
that  region.  The  condensations  not  only  augment,  as  the  contraction  goes  on,  in 
directions  towards  the  equatorial  plane,  but  also  towards  the  centre ;  since  all  the 
stars,  except  those  moving  in  the  plane  of  the  equator,  tend  towards  the  centre,  in 
their  orbital  motions.  The  greatest  density  will  be  attained  at  the  centre ;  and  at 
the  epoch  when  the  polar  stars  have  reached  its  vicinity. 

Possible  collisions.  In  all  the  movements  of  the  stars  of  the  supposed 
cluster,  the  only  collisions  that  could  directly  ensue  would  be  in  those  cases  in 
which  two  stars  set  out  from  two  points  of  a  meridian,  at  the  same  distance  from 
the  equator,  at  the  same  epoch.  Each  of  the  two  stars  should,  at  the  end  of  a 
quarter  revolution,  reach  the  plane  of  the  equator  at  the  same  point.  Such  exact 
coincidences  of  position  and  date  of  origin  should,  however,  rarely  occur.  But 
frequent  close  approximations  of  two  or  more  stars  may  well  occur,  and  eventuate 
in  the  formation  of  double  or  triple  stars,  revolving  around  each  other. 

Spheroidal  clusters.  We  are  led,  by  the  theoretical  views  that  have  now  been 
presented,  to  regard  the  oval  nebulae,  or  spheroidal  clusters,  seen  in  the  heavens, 
as  original  globular  clutters  in  some  of  their  different  stages  of  spheroidal  condensation. 
Upon  this  hypothesis  they  should  in  general  be  more  condensed  and  more  difficult 
of  resolution  than  existing  globular  clusters  (as  is  observed  to  be  the  case).  The 
amount  of  oblateness,  with  the  attendant  condensation,  would  be  an  index  of  the 
age  of  the  cluster.  Present  globular  clusters  would  be  just  at  the  beginning  of 
their  first,  or  of  a  new  grand  cycle ;  and  destined  in  future  ages  to  pass  through 
the  continued  series  of  spheroidal  forms  that  we  have  signalized. 

Inequalities  of  brightness  in  different  parts  of  spheroidal  clusters.  Instances  of 
such  inequalities,  not  resulting  from  a  mere  central  condensation,  are  observable  in 
Fig.  6,  Plate  IV,  and  in  Fig.  9,  Plate  V.  They  may  be  attributed  to  inequalities 
of  density  existing  at  certain  stages  of  the  contraction  of  the  original  globular 
cluster.  For  example,  at  the  end  of  a  half  revolution  of  the  polar  stars  they  would 
be  condensed  about  the  contracted  poles,  which  would  have  passed  to  the  opposite 
sides  of  the  equatorial  plane,  and  the  other  stars  of  each  original  spherical  layer 
would  be  condensed  in  the  resulting  spheroidal  layer,  but  in  a  decreasing  degree 
towards  the  equator.  This  theoretical  condition  answers  to  the  dumb-bell  nebula 
(Fig.  9,  Plate  V).  The  difierences  observable  in  the  distribution  of  the  light  on 
opposite  sides  of  the  equatorial  plane  (or  of  the  larger  axis  of  the  faint  elliptic  out- 
line) are  such  as  might  result  from  a  want  of  entire  correspondence  in  the  epochs 
4>f  the  initial  orbital  motions  of  the  stars  on  opposite  sides  of  that  plane.  Fig.  6, 


ORIGIN   OF  SIDEREAL  SYSTEMS.  441 

Plate  IV,  answers  to  a  similar  period  in  the  structural  history  of  a  cluster ;  but 
the  inner  layers  have  experienced  a  more  marked  condensation  towards  the  centre. 
This  would  be  the  result  if  the  initial  velocities  of  the  stars  of  these  layers  were, 
from  some  special  cause,  materially  less  than  the  normal  values.  Under  the  same 
circumstances  these  layers  would  be  more  nearly  spherical  in  their  form  than  in 
the  normal  type  above  considered. 

Case  II.  A  simuUaneous  disruption  of  the  nebulous  body  along  a  limited  num- 
ber of  meridians.  We  wih1  now  suppose  the  disruptive  evolution  of  the  starry 
masses  to  be  confined  to  certain  meridians,  and  the  regions  contiguous  to  them  on 
either  side,  and  enquire  into  the  subsequent  form  and  internal  condition  of  the 
cluster.  It  will  readily  be  seen  that,  since  the  initial  velocities  and  the  periods  of 
revolution  decrease  from  the  equatorial  to  the  polar  stars,  the  stars  proceeding  from 
any  one  of  these  meridians,  will,  as  they  follow  their  natural  orbits,  take  on  collect- 
ively a  spiral  arrangement.  This  will  be  best  seen  when  viewed  perpendicularly  to 
the  equatorial  plane.  At  the  close  of  a  half  revolution  of  the  polar  stars,  the 
spiral,  thus  viewed,  would  occupy  the  second  quadrant  of  revolution  of  the  entire 
set  of  stars  considered.  As  the  revolution  proceeded,  the  angular  extent  of  the 
spiral  would  continually  increase.  The  stars  proceeding  from  the  meridian  con- 
tiguous to  that  first  supposed,  would  form  a  similar  spiral  contiguoas  to  that  just 
considered ;  and  the  entire  collection  of  stars  setting  out  from  the  one  meridional 
region  of  disruption  would  form  a  spiral  band,  increasing  in  width  from  its  inner 
to  its  outer  end.  The  similarly  situated  stars  of  the  successive  layers,  proceeding 
inwards,  would  form  shorter  spirals,  that  would  be  combined  with  the  others  and 
add  to  the  width  of  the  spiral  band.  A  similar  spiral  band  would  result  from  each 
of  the  other  collections  of  detached  stars. 

Spiral  nebu/ce.  According  to  what  has  just  been  shown,  the  spiral  formation  is 
a  natural  consequence  of  a  cotemporaneous  evolution  of  stars  from  various  points 
of  the  same  meridian.  The  spiral  arrangement  of  stars  should  therefore  exist  in 
every  cluster,  and  be  more  or  less  discernible,  unless  the  disruption  was  general  and 
nearly  simultaneous  throughout  the  original  nebulous  mass.  Accordingly  spiral 
lines  and  fringes  of  stars  are  in  many  instances  observable  on  the  borders  of 
spheroidal  and  irregular  clusters. 

The  theory  of  the  origin  of  the  true  spiral  nebuke  has  been  sufficiently  indicated. 
The  length  of  the  spiral  coils  in  Fig.  10,  Plate  Y,  indicates  that  the  equatorial  stars 
of  the  cluster  have  completed  three-quarters  of  a  revolution,  and  the  polar  stars  ono 
revolution  and  a  half.  The  secondary  condensation,  on  the  right  of  the  figure, 
may  be  ascribed  to  the  circumstance  of  the  stars  that  set  out  from  points  on  the 
meridian  near  the  equator,  being  at  the  present  epoch  in  the  vicinity  of  one  of 
the  nodes ;  three-quarters  of  a  revolution  having  been  completed.  The  nebula 
seems  to  consist  of  two  spiral  bands  or  coils,  made  up  of  an  indefinite  number  of 
spiral  filaments,  or  smaller  bands.  The  line  of  sight  is  probably  nearly  perpendicu- 
lar to  the  line  of  the  two  centres  of  condensation,  and  oblique  to  the  equatorial 
plane.  The  great  comparative  dispersion  of  the  filaments  of  the  lower  coil  may  be 
in  part  attributable  to  the  detachment  of  stars  beginning  on  one  meridian,  and  ex- 
tending gradually  around  to  successive  meridians. 

Case  III.  An  irregular  disruption.  If  irregular  deviations  from  the  normal 
type  of  evolution  that  has  been  considered  occur,  the  result  should  be  the  forma- 
tion of  irregular  clusters  differing  more  or  less  from  true  globular  or  spheroidal 
clusters.  By  reason  of  the  want  of  correspondence  in  the  epochs  of  detachment 
on  difterent  meridians  of  the  nebulous  body,  such  clusters  should  be  less  condensed, 
and  with  less  regularity  than  the  regular  clusters. 

Case  IV.  A  disruption  beginning  at  the  equator,  and  extending  graduaUy  towards 
the  pules.  When  this  deviation  from  the  normal  type  occurs,  the  obvious  result 
will  be  that  the  arrival  of  the  stars  at  the  equatorial  plane  will  be  delayed,  in 
proportion  to  their  angular  distance  from  it,  at  tlie  outset ;  and  therefore  that  the 
contraction  of  each  of  the  original  spherical  layers  will  take  place  more  slowiy. 
The  condensation  towards  the  equatorial  plane  will  also  go  on  more  slowly.  The 
law  of  retardation  of  the  dates  of  initial  movement,  for  stars  at  increasing  distances 
from  the  equator,  may  theoretically  be  such  as  to  determine  any  known  law  of 
decrease  of  density  along  each  spheroidal  layer,  from  the  equator  to  the  axis. 

System  of  the  Milky  Way.  It  is  accordingly  conceivable  that  the  observed  law 
of  decrease  of  the  density  of  the  system  of  the  milky  way,  as  the  distance  from 
its  principal  plane  increases,  may  have  resulted  from  the  operative  cause  just  con- 


442  APPENDIX. 

sidered.  If  the  same  law  of  evolution  prevailed  cotemporaneousiy,  or  approxi- 
mately so,  throughout  the  mass  of  the  nebulous  body  from  which  this  system  is 
supposed  to  have  been  derived,  the  result  would  be  the  formation  of  spheroidal 
layers  of  stars  in  which  the  density  would  vary  according  to  a  common  law.  The 
density  would  therefore  decrease  outwardly  from  the  principal  plane  of  the  milky 
way  at  the  same  rate  for  stars  at  all  distances.  Struve  found  this  to  be  nearly 
true  for  all  stars  beyond  the  6th  or  7th  magnitude. 

Motion  of  revolution  of  the  sun.  Since  the  sun  is  now  near  the  equatorial  plane, 
and  moving-  away  from  it  (Art.  474),  we  must  suppose  that  it  has  at  least  com- 
pleted either  one-quarter  or  three-quarters  of  a  revolution  in  its  vast  orbit.  The 
position  of  the  centre  of  the  system  is  too  imperfectly  known  (Art.  478)  to  make 
it  possible  to  determine  with  certainty  which  of  the  two  nodes  it  is  now  leaving1. 
Taking  the  average  velocity  of  the  sun  at  4|  miles  per  second,  as  given  in  the 
text,  and  supposing  the  distance  of  the  centre  of  the  system  to  be  equal  to  the 
exterior  limit  of  stars  of  the  6th  magnitude,  the  period  of  revolution  of  the  sun 
should  be  35  millions  of  years.  MMler's  estimate  of  the  distance  of  the  centre  of 
the  system  of  the  milky  way  places  it  a  little  beyond  the  exterior  limit  of  stars  of 
the  5th  magnitude.  The  above  larger  estimate  answers  nearly  to  Struve's  deter- 
mination of  the  sun's  orbital  velocity  (Art.  448). 

Particular  features  of  the  Milky  Way.  The  variable  breadth  of  the  belt  of  the 
milky  way,  its  bifurcation,  and  alternations  of  bright  and  dark  patches,  may  have 
proceeded  from  a  want  of  correspondence  in  the  dates  of  evolution  of  stars  on  dif- 
ferent meridians  of  the  original  nebulous  mass,  as  well  as  abnormal  interruptions 
of  the  process  of  separation  at  certain  parts  of  particular  meridians.  Thus,  if  on  a 
certain  meridian,  the  separation  into  stars  should  not  have  occurred  for  a  certain 
distance  from  the  equator,  on  either  side,  it  would  follow  that  just  before  or  after 
•J-  of  a  revolution  of  the  stars  that  set  out  from  the  points  nearest  the  equator,  the 
stars  from  greater  latitudes  would  be  concentrated  upon  two  points  at  a  short  dis- 
tance from  the  equator,  on  either  side.  If  the  same  initial  circumstances  prevailed 
over  a  series  of  meridians,  there  would  be  formed  two  bands  of  greatest  condensa- 
tion at  a  certain  distance  from  the  equator,  on  opposite  sides.  The  same  bands 
would  manifest  themselves  before  and  after  f  of  a  revolution. 

Primary  condition,  and  subsequent  processes  of  change,  of  fragments  disunited  from 
the  primitive  nebulous  mass.  Since  the  parts  of  any  such  fragment  unequally  dis- 
tant from  the  axis  must  have  had,  at  the  time  of  separation,  unequal  velocities  of 
rotation,  it  must  have  taken  up  a  rotation  about  an  axis  of  its  own,  and  tended  to 
assume  the  form  of  a  sphere,  or  spheroid  somewhat  flattened  at  the  poles.  It 
should  therefore,  at  some  subsequent  date,  have  broken  up  into  a  cluster  of  stars, 
or  into  a  planetary  system  revolving  around  a  central  sun.  If  the  mass  detached 
should  have  been  of  comparatively  great  extent,  it  may  have  separated  into  a  com- 
bination of  stars  and  clusters  of  stars,  as  in  the  Magellanic  Clouds  ;  or  into  irregu- 
lar beds  of  stars,  as  in  the  irregular  nebulas. 

The  concentration  attending  the  formation  and  subsequent  contraction  of  such 
systems  should  have  occasioned  a  vacuity  of  stars  in  the  spaces  contiguous  to 
them. 

Irresolvable  nebula.  If,  as  we  have  already  been  led  to  suppose,  the  process  of 
evolution  of  the  system  of  the  milky  way  from  a  primitive  nebulous  mass,  extend- 
ed gradually  from  the  equator  towards  the  poles,  the  vast  nebulous  mass  left 
detached  at  the  outer  polar  regions,  and  subject  to  peculiar  conditions,  may  have 
become  disunited  into  large  masses,  from  which  clusters  have  been  derived  that 
are  now  at  an  earlier  stage  of  development,  and  at  a  greater  distance  than  the 
telescopic  stars  and  clusters. 

Annular  wbulce,  which  are  the  rarest  objects  in  the  heavens,  may  have  resulted 
from  the  matter  of  the  polar  regions  of  a  nebulous  body  being  mostly  drawn  to 
surrounding  points  of  condensation,  or  not  having  yet  condensed  into  true  stars, 
or  into  stars  comparatively  minute. 

Planetary  nelmke  probably  belong  to  the  same  type  of  development  as  annular 
nebulae;  since  some  of  them  have  been  resolved  into  annular  nebulae.  Their 
equable  light  may  result  from  the  process  of  star-formation  having  penetrated 
only  a  certain  depth  into  the  original  nebulous  mass. 

^  General  Considerations.  We  may  conclude  from  the  previous  theoretical  discus- 
sion, that  if  we  assume  all  systems  of  stars  to  have  been  derived,  by  separation, 
from  rotating  nebulous  bodies  of  vast  extent,  according  to  one  or  the  other  of  a 


ORIGIN  OF  SIDEREAL  SYSTEMS.  443 

certain  small  number  of  types  of  evolution,  the  forms  and  internal  conditions  that 
would  be  inevitably  passed  through,  in  the  progress  of  ages,  would  be  the  coun- 
terparts of  the  various  forms  and  apparent  structural  conditions  of  the  clusters 
and  nebulae  actually  observed.  "We  may  suppose,  it  is  true,  in  explanation  of  the 
single  case  of  the  spheroidal  clusters,  that  the  rotating  bodies  took  on  a  decidedly 
spheroidal  form  before  disintegration,  and  that  the  derived  clusters  have  now, 
sensibly,  their  original  form.  Upon  this  view  we  must  still  admit  that  these  clus- 
ters are  destined,  in  the  lapse  of  future  ages,  to  pass  through  changes  similar  to 
those  we  have  deduced  for  globular  clusters.  But  spiral,  and  some  other  forms 
of  nebulae,  cannot  but  have  passed  through  vast  ages  of  development,  and  in  the 
light  of  this  indication  of  the  age  of  the  heavens,  it  seems  improbable  that  sphe- 
roidal clusters  should  be  of  comparatively  recent  origin. 

As  to  the  real  nature  of  the  process  of  separation  of  starry  fragments  from 
rotating  nebulous  masses,  we  are  led,  on  physical  grounds,  to  conceive  that  it 
must  have  consisted  in  a  concentration  upon  special  points  of  the  mass  at  certain 
favorable  epochs,  rather  than  in  a  violent  separation.  It  would  seem  that  such 
a  process  beginning  at  any  part  of  the  nebulous  body,  should  tend  to  extend  in- 
definitely through  it.  But  it  is  to  be  observed  that  the  propagation  of  a  force 
through  such  a  mass  would  of  itself  require  a  vast  period  of  time.  It  is  possible 
that  the  want  of  correspondence  in  the  epochs  of  separation,  in  different  parts  of 
the  body,  we  have  inferred  existed  in  the  development  of  the  system  of  the  milky 
way,  and  to  a  less  extent  also  in  spiral  nebulae,  may  have  resulted  from  the  far 
greater  comparative  size,  in  these  instances,  of  the  original  nebulous 


TABLE  I. 
Latitudes  and  Longitudes  of  Places. 


Names  of  Places. 

Latitude. 

Longitude  from 
Greenwich  in  Arc. 

!    Long,  from 
Greenwich  in 
Time. 

Longitude  from 
Washington 
in  Time. 

Abo,  Obs 

60"  26'  56".8  N. 
42  39  50  .0  N. 
53  32  45  .3  N. 
42  16  48  .0  N 
46  11  27  .6  N. 
37  58  20  .0  N 
39  17  47  .8  N 
52  30  16  .7  N 
42  21  27  .6  N 
42  22  48  .6  N 
33  56  3  .0  S 
55  58  41  S 
32  46  33  .1  N 
55  40  53  .0  N 
58  22  47  .1  N 
53  23  13  .0  N. 
55  57  23  .2  N 
29  18  17  .3  N 
50  56  5  .2  N. 
51  31  47  .9  N 
51  28  38  .0  N 
53  33  5  .0  N. 
43  3  16  .5  N. 
55  47  23  .1  N. 
54  42  50  .7  N. 
53  25  28  .0  N. 
51  31  29  .8  N. 
13  4  8  .1  N. 
40  24  27  .7  N. 
43  17  49  .0  N. 
45  28  0  .7  N. 
55  45  19  .8  N. 
44  21  3  .9  N. 
40  51  46  .6  N. 
41  18  36  .5  N. 
29  57  45  .0  N. 

22°  17'  11".4  E. 
73     44   39  .0  W. 
9     56   32  .3  E. 
83     43      3  .0  W. 
123     49   31  .7W. 
23     43   47  .8  E. 
76     36   38  .6W. 
13     23   52  .8  E. 
71       3   30  .OW. 
71       7    24  .9  W. 
18     28   45  .0  E 
67     10   53  .OW 
79     55   37  .6W 
12     34   57  .0  E 
26     43    23  .4  E 
6     20   30  .OW 
3     10   45  .OW 
101     33    33  .0  W. 
10     43   54  .9  E 
9     56   31  .5   E. 
0       0      0  .0 
9     58   23  .4  E. 
75     24   16  .8W. 
49       6   36  .6  E. 
20     30     5  .4  E. 
2     54  40  .5W. 
0       9    16  .5W. 
80     14   15  .0  E. 
3     41    21  .OW. 
5     22    H  .8  E. 
9     11    39  .6  E. 
37     34   14  .4  E. 
68     13    15  .5W. 
14     14  42  .9  E. 
72     55    30  .OW. 
90       6   49  .OW. 
74       0     3  .IW. 
13     21      2  .6  E. 
151       1    33  .7  E. 

h.     m.     s. 
—1.29     8.8 
+  4  54  56.6 
—0  39  46.1 
+  5  34  52.2 
+  8  15  18.1 
—1  34  55.2 
+  5     6  26.6 
—0  53  35.5 
+  4  44  14.0 
+  4  44  29.7 
—  1  13  55.0 
+  4  28  43.5 
+  5  19  42.5 
—  0  50  19.8 
—1  46  53.6 
+  0  25  22.0 
+  0  12  43.0 
+  6  46  14.2 
—  0  42  55.7 
—  0  39  46.1 
0     0     0.0 
—  0  39  53.6 
+  5     1  37.1 
—3  16  26.3 
—3  22     0.4 
+  0  11  38.7 
+  0     0  37.1 
—5  20  57.0 
+  0  14  45.4 
—  0  21  29.0 
—  0  36  46.6 
—2  30  17.0 
+  4  32  53.3 
—  0  56  58.9 
+  4  51  42.0 
+  6     0  27.0 
+  4  56     0.2 
—  0  53  24.2 
—3  39  31.4 
—  0     9  20.6 
—2     1   13.5 
—5  15  44.8 
+  9  57  56.0 
+  4  58  38.3 
—2     1  18.7 
+  4  44  49.0 
+  2  52  14.6 
—  0  49  54.7 
+  8     9  32.5 
+  4  42  33.0 
+  5  24  20.9 
—  1  12  14.8 
—  1     5  32.5 
+  5     8  11.2 

h.     m.      s. 
—  6  37   20.0 
—  0  13  12.6 
—  5  47  57.4 
+   0  26  41.0 
+   3     7     6.9 
—  6  43     6.4 
—  0     1  45.4 
—  61  46.7 
—  0  23  57  2 
—  0  23  41.5 
—  6  22     6.2 
—  0  39  27.7 
-f   0  11   31.x 
—  5  58  31.0 
—  6  55     4.8 
—  4  42  49.2 
—  4  55  28.2 
-f   1  38     3.0 
—  5  51     6.9 
—  5  47  57.3 
—  58  11.2 
—  5  48     4.8 
—  06  34.1 
—  8  24  37.5 
—  6  30  11.2 
—  4  56  32.5 
—  57  34.1 
—10  29     8.2 
—  4  53  25.8 
—  5  29  40.2 
—  5  44  57.8 
—  7  38  28.2 
—  0  35  17.9 
—  65  10.1 
—  0  16  29.2 
+   0  52  15.8 
—  0  12  11.0 
—  61  35.4 
—  8  47  42.6 
—  5  17  31.8 
—  7     9  24.7 
—  07  33.6 
+   4  49  45.0 
-09  32.9 
—  79  29.9 
—  0  23  22.2 
—  2  15  56.6 
—  5  58     5.9 
+   3     1  21.3 
—  0  25  38.2 
+    0  16     9.7 
—  6  20  26.0 
—  6  13  43.7 
—  0     0     0.0 

Albany,  Obs  ....         ... 

Altona  Obs 

Arm  Arbor,  Obs  
Astor  Point,  Oregon.   .  . 
Athens,  Ob-->  

Baltimore,   Wash.  Mi... 
Berlin,  Obs  

Boston,  State  House.  .  . 
Cambridge,  Obs. 

Cape  of  Good  Hope,  Obs 
Cape  Horn 

Charleston,  St.  MicVs  Ch 
Copenhagen,  Obs  

Dublin    Obs 

Edinburgh,  Obs  

Galveston    Oath 

G-otha,  Obs.   

Gottin°'eu    Obs 

Greenwich,  Obs.     .  .   . 

Hamburg,  Obs  

Hamilton  Coll.,  Obs  
Kazan,   Obs  

K.6nicrsber°'   Obs         . 

Liverpool,  LasseU  Obs.  .  . 
London,    Obs 

Madras,  Obs  

Madrid    Obs 

Marseilles,   Obs  

Milan,  Obs  
Moscow,  Obs  

Mount  Desert,  Maine  .  .  . 
Naples,  Obs  ... 

New  Haven,  She/.  Obs.  . 
New  Orleans  

New  York,  City  Hall.  .  . 
Palermo,  Obs  
Paramatta,  Obs  

40  42  43  .2  N. 
38  6  44  .0  N. 
33  48  49  .8  S. 
48  50  13  .2  N. 
59  56  29  .7  N. 
39  57  7  5  N 

Paris,   Obs     .  . 

2     20     9  .4  E. 
30     18    22  .2  E. 
75       9    23  .4W. 
149     28    55  .OW. 
74     39   34  .3  W. 
30     19   39  .9  E. 
71     12    15  .OW. 
43       3   39  .OW. 
12     28   40  .5  E. 
122     23    19  .4W. 
70     38    14  .5W. 
81       5    14  .3  W. 
18       3   42  .0  E. 
16     23      7  .9  E. 
77       2   48  .OW. 

St.  Petersburg,  Obs  

Point  Venus,  Otaheite  .  . 
Princeton,  Seminary... 
Pulkowa,   Obs  

17  29  21  .0  S. 
40  20  40  .0  N. 
59  46  18  .7  N. 
46  48  30  .0  N. 
22  53  51  .0  S. 
41  53  52  .2  N. 
37  47  59  .2  N. 
33  26  25  .4  S. 
32  4  53  .4  N. 
59  20  31  .0  N. 
48  12  35  .5  N. 
38  53  39  .3  N. 

Quebec   Obs        

Rio  de  Janeiro,  Obs.  .  .  . 
Rome   Obs     

San  Francisco,  Tel.  Hill.  . 
Santiago  de  Chile,  Obs.  . 
Savannah  Exch        .  .  . 

Stockholm,  Obs  

Vienna,  Obs  

Washiuo'ton.  Obs 

TABLE  II. 

Elements  of  the  Planetary  Orbits. 
Epoch,  January  1,  1850,  mean  noon  at  Paris. 


Planet's  Name. 


Mercury 

Venus 

Earth   

Mars 

Jupiter 

Saturn  

Uranus 

Neptune 


Inclination  to  L       v 
the  Ecliptic.  S*c.Var 


7°  0'  8' 
3  23  31 


51  5 

18  40 

29  28 

46  30 

46  59 


+  4  .5 

—  2  .4 
—23 
—15 
+  3 


Longitude  of    L      ,7 
AscendingNode.  Sec'Var- 


46°  33'  9"  71'  4' 
75  19  52  54  49 


48 
98 

112 
73 

130 


23  43 

54  20 

21  44 

14  14 

6  52 


46  39 

57  14 

51  10 

23  39 


Longitude  of 
Perihelion. 


75C 
129 
100 
333 

11 

90 
168 

47 


7'  14" 
27    14 
21    21 
17    54 
54   53 

6  12 
16  45 
14  37 


Sec.  Van 


93'  11' 

82  26 

102  50 

110  24 

94  49 

115  55 

87  32 


Mercury 
Venus  . . 
Earth. . . 
Mars  . . . 
Jupiter  . 
Saturn  . . 
Uranus. . 
Neptune. 


M.  Distance 
from  Sun,  or 
Semi-Axis. 


0.3870987 
0.7233322 
1.0000000 
1.523691 
5.202798 
9.538852 
19.182639 
30.03697 


Mean  Distance 

from  Sun  in 

Miles. 


35,353,000 

66,060,000 

91,328,000 

139,156,000 

475,161,000 

871,164,000 

1,751,912,000 

2,743,216,000 


Eccentricity 

0.2056048 
0.0068433 
0.0167711 
0.0932611 
0.0482388 
0.0559956 
0.0465775 
0.0087195 


Sec.  Variation. 


+  0.000020294 
—  0.000053843 
—0.000042582 
+  0.000095284 
+  0.000159350 
—0.000812402 
—0.000025072 


Mercury 
Venus. . . 
Earth . . . 
Mars  . . . 
Jupiter  . 
Saturn  . . 
Uranus  . 
Neptune 


Mean  Longitude 
at  the  Epoch. 


327°  15'  20" 
245     33    15 
46    43  .5 


280 
83 

160 
14 
28 

335 


40  31 

1  20 

50  41 

26  41 

8  58 


M.  Sidereal  Period 
in  Mean  Solar 
Days. 

Motion  In  M*an 
Long,  in  1  yr.  of 
865  days. 

Mean  Daily  Mo- 
tion in 
Longitude. 

d. 
87.9692580 
224.7007869 
365.2563744 
686.9796458 
4332.5848212 
10759.2198174 
30686.8208296 
60126.72 

53°  43'     3" 
224    47    30 
359     45    41 
191     17      9 
30     20    32 
12     13    36 
4     17    45 
2     11    58 

4°     5'  32".6 
1     36      7  .8 
0     59      8  .33 
0     31    26  .7 
0       4   59  .3 
0       2      0  .6 
0       0   42  .4 
0       0    21  .7 

TABLE  III. — Elements  of  Moon's  Orbit. 
Epoch,  January  1,  1801,  mean  noon  at  Paris. 


Mean  inclination  of  orbit 5°     8'  40" 

Mean  longitude  of  node  at  epoch 13     53   17.  7 

Mean  longitude  of  perigee  at  epoch 266     10     7  .5 

Mean  longitude  of  moon  at  epoch 118     12      8.3 

Mean  distance  from  earth 60.2665590 

Eccentricity 0.05490807 

d.     h.    m.    s.  d. 

Mean  sidereal  revolution 27     7  43  11.5  =  27.32166142 

Mean  tropical          "         27     7  43     4.7  =  27.32158242 

Mean  synodical       "         291244     2.9  =  29.53058872 

Mean  anomalistic   "         27  13  18  37.4  =  27.55459950 

Meannodical          »         27     5     536.0  =  27.21222222 

d,  d. 

Mean  revolution  of  nodes:  sidereal =  6793.432  ;  tropical =  6798.33557 

Mean  revolution  of  perigee:  sidereal.. .  =  3232.57534;  tropical. .  =  3231.4751 


TABLE  IV.  3 

Diameters,  Volumes^  Masses,  etc.,  of  Sun,  Moon,  and  Planets. 


Mercury  . 
Venus  . . 
Earth . . . 
Mar*  . .  . 
Jupiter  . 
Saturn  . 
Uranus. . 
Neptune. 
Sim  . . . 
Moon  . 


Apparent  Diameter. 


Least. 


.-]  1 


4".5 
9  .7 

4  .1 

30  .8 

14  .6 

3  .5 

2  .6 

32 


28    48 


At  Moan 
Distance. 


6".7 
17  .0 

11  .1 

37  .2 

16  .1 

3  .9 

2  .7 


32' 
31 


Greatest. 


12".9 

66  .3 

30  .1 

50  .6 

2' >  .3 

4  .3 

2  .9 


3  .4*  32'  36 
7  .0    33    32 


Equatorial 
Di.im  ter. 


0.3732 
0.9525 
1.0000 
0.6201 
11.1401 
9.0621 
4.1864 
4.5383 
107.263 
0.2725 


Equatorial 

Diameter  in 

Miles. 


2,958 

7,549 

7,925.6 

4,915 

88,294 

71,823 

33,124 

35,910 

850,123 

2,160f 


Volume. 


0.0518 
0.8686 
1.0000 
0.2345 
1303.91 
667.54 
73.369 
93.470 
1,240,285.0 

0.0203 


Mercury 
Venus. . , 
Earth  . .  , 
Mars.... 
Jupiter . . 
Saturn  .  . 
Uranus . . 
Neptune . 

Sun 

Moon  . . 


Mass. 


Density. 


2.020 
0.903 
1.000 
0.447 
0.229 
0.134 
0.178 
0.179 
0.253 
0.602 


Gravity  at 
Equator. 


0.751 
0.865 
1.000 
0.273 
2.410 
1.089 
0.746 
0.812 
27.292 
0.164 


Sidereal 

llotatiim. 


h.     m.    8. 
24     5   28 
23  21   24 

23  56     4 

24  37  20 
9  55  26 

10  29  17 


h.  m 
4  29* 
7  43 


Light  and 
Heat. 


6.680 

1.911 

1.000 

.431 

.037 

.011 

.003 

.001 


TABLE  V. 
Elements  of  the  Retrograde  Motion  of  the  Planets. 


P  anets. 

Arc  of  Retro- 
gradation. 

Duration  of  Retro- 
gradation. 

Elongation  at  the 
Stations. 

Synodic 
Revolution. 

d.    h.            d.    h. 

d. 

Mercury  . 

9°  22'  to  15°  44' 

23   12  to     21   12 

14°  49'  to    20°  51' 

115.8775 

Veuus.  .  .  . 

14     35  to  17    12 

40  21  to    43  12 

27     40  to    29    41 

583.9214 

Mars  10       6  to  19    35 

60  18  to    80  15 

128     44  to  146     37 

779.9364 

Jupiter.  .  . 

9     51    to     9    59  116  18  to  122  12 

113     3-5  to  116     42 

398.8841 

Saturn  .  .  . 

6     41   to     6    55  138  18  to  135     9 

107     25  to  110     46 

378.0919 

Uranus  .  . 

3     36 

151 

103     30 

369.6563 

Neptune  . 

367.4888 

*  The  value  32'  0"  used  in  the  text  (269)  is  the  above  value  corrected  for  irra- 
diation, and  agrees  with  that  obtained  from  observations  of  the  transits  of  Mercury 
over  the  sun's  disc. 

f  The  value  of  the  moon's  diameter  (2161.6  m.)  obtained  in  the  text,  is  probably 
about  2  miles  too  large,  as,  according  to  Airy,  a  correction  of  2"  should  be  applied 
to  the  moon's  measured  diameter  for  the  effect  of  irradiation,  and  1"  answers  to 
about  1  mile. 

J  This  is  the  mass  of  Mercury  adopted  by  Le  Verrier.  Many  astronomers  still 
retain  Encke's  determination,  which  is  TsFsTaT-  This  gives  for  the  density  of 
Mercury  1.246. 

§  This  is  Faye's  recent  determination.  According  to  Carrington,  the  most  pro- 
bable value  is  24d.  23h.  30m.  Sporer's  value  is  25d.  5h.  37m. 


TABLE  VI 


Ekments  of  the  Orbits  of  the  Satellites. 

The  distances  are  expressed  in  equatorial  radii  of  the  primaries. 
I.  Satellites  of  Jupiter. 


Satellites. 

Mean 
Distance. 

Sidereal  Revolution. 

Inclination  to 
Orbit  of 
Jupiter. 

Epoch 
of  El'ts. 

Mass  ;  that  of 
Jupiter  —  1. 

1  

6.04853 

d.    h.   m.       s. 
1    18  27   33.506 

3°      5'   30" 

Jan.  1, 

0.000017328 

2  

9.62347 

3  13  14  36.393 

Variable. 

1801. 

0.000023235 

8  

15.35024 

7     3  42  33.362 

Variable. 

a.  T. 

0.000088497 

4 

26  99835 

16  16  31  49.702 

2     68   48 

0.000042659 

II.  Satellites  of  Saturn. 


Name  and  Order. 

Mean 
Distance. 

Sidereal 
Revolution. 

M.  Long, 
at  the  Epoch. 

Eccentricity  and 
Perisaturnium. 

Epoch 
of  El'ts. 

1.  Mimas  .... 
2.  Enceladus.  . 
3.  Tethys  
4.  Dione  
5.  Rhea  
6  Titan 

3.3607 
4.3125 
5.3396 
6.8398 
9.5528 
22  1450 

d.    h.    m.     s. 
0  22  37  22.9 
1     8  53     6.7 
1  21  18  25.7 
2  17  41     8.9 
4  12  25  10.8 
15  22  41  25.2 

256°  58'  48" 
67     41    36 
313     43   48 
327     40   48 
353     44      0 
137     21    24 

0.04(7);  54°(?) 
0.02  (?);  42    (?) 
0.02  (?);   95    (?) 
0  0293-  256  38' 

1790.0 
18360 
Ditto. 
Ditto. 
Ditto. 
1830.0 

7.  Hyperion  .  . 
8.  lapetus  .... 

28 
64.3590 

21     7     7 
79     7  53  40.4 

269     37   48 

Apsides  of  Titan  have  a 
motion  in  long,   of  30' 
28"  per  an. 

1790.0 

The  longitudes  are  reckoned  in  the  plane  of  the  ring  from  its  descending  node 
with  the  ecliptic.  The  first  seven  satellites  move  in  or  very  nearly  in  its  plane  ; 
the  orbit  of  the  8th  lies  about  half  way  between  the  plane  of  the  ring  and  that  of 
the  planet's  orbit. 

III.  Satellites  of  Uranus. 


Satellites. 

M.  Dis- 
tance. 

Sidereal 
Revolution. 

Passage  through 
Asc.  Node,  G.  T. 

Inclination  to  Ecliptic. 

1.  Ariel  

7.44 
10.37 
13.12 
17.01 
19.85 
22.75 
45.51 
91.01 

d.  h.    m. 
2  12  28 
4     3  27 
5  21  25 
8  16  56 
10  23     3 
13  11     7 
38     1  48 
107  16  39 

d.  h.  m. 
1787,  Feb.  16  0  10 

1787,  Jan.     7  0  28 

The   orbits    are   in- 
clined  at  an  angle  of 
about  79°  to  the  eclip- 
tic in  a  plane  whose  as- 
cending' node  is  in  long. 
165°    30'    (Equinox   of 
1798).    Their  motion  is 
retrograde.  Their  orbits 
are  nearly  circular. 

2    Umbriel 

3  

4    Titania 

5  

6.  Oberoii  

7      

8  

IV.    Satellite  0     Neptune.    Period,  5.877  d.;  M.  Distance,  12  radii  of  Planet. 
TABLE  VII.     Saturn's  Ring. 


Outer  diameter  of  outer  ring 40".095 1 69,341  miles. 

Inner  diameter  of  outer  ring 149,060  " 

Breadth  of  outer  ring 10,149  " 

Breadth  of  inner  ring 16,484  " 

Interval  between  rings 1,723  " 

Breadth  of  double  ring 28,356  " 

Distance  of  ring  from  planet 18,327  " 

Thickness  of  the  rings  not  exceeding. 210  " 


TABLE  H  (a). 

The  Planetoids. 

Names,  particulars  of  discovery,  mean  distances,  etc. 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 
Distance. 

Sidereal 
Period. 
Yrs. 

Eccentricity. 

1. 

Ceres  

1801,  Jan.  1. 

Piazzi. 

2.7660 

4.600 

0.08024 

2. 

Pallas  

1802,  March  28. 

Olbers. 

2.7700 

4.610 

0.23969 

3. 

4. 

Juno  
Vesta  

1804,  Sept.  1. 
1  807,  March  29. 

Harding. 
Olbers. 

2.6687 
2.3607 

4.362 
3.627 

0.25590 
0.09012 

5. 
6 

Astrsea  .  
Hebe 

1845,  Dec.  8. 
1847,  July  1. 

Hencko. 
Hencke. 

2.5775 
24*>54 

4.136 
3777 

0.18999 
020115 

7 

Iris.  ..  . 

"    Aug.  13. 

Hind. 

2.3862 

3.686 

0.23125 

8. 

"    Oct.  18. 

Hind. 

2.2014 

3.266 

0.15670 

9 

Metis 

1848,  April  25. 

Graham. 

2  3862 

3.686 

0  12320 

10 

Hegeia  

1849,  April  12. 

De  Gasparis. 

3.1494 

5.589 

0.10056 

11. 
12. 
13 

Parthenope  . 
Victoria  
Eo'eria  .  . 

1850,  May  11. 
"     Sept.  13. 
"    Nov.  2. 

De  Gasparis. 
Hind. 
De  Gasparis. 

2.4526 
2.3344 
2  5756 

3.841 
5.567 
4.133 

0.09888 
0.21890 
008775 

14. 

Irene  

1851,  May  19. 

Hind. 

2  5895 

4.167 

0.16595 

15. 

16. 
17 

Eunomia  .  .  . 

Psyche  
Thetis  .  . 

"    July  29. 

1852,  March  17. 
"    April  17. 

De  Gasparis. 

De  Gasparis. 
Luther 

2.6429 

2.9263 
2  4737 

4.297 

5.006 
3  890 

0.18801 

0.13575 
0  12686 

18. 
19. 
20. 

21. 
22. 
23 

Melpomene  . 
Fortuna.  .  .  . 
Massilia  

Lutetia  
Calliope.  .  .  . 
Thalia  

"     June  '24. 
"     Aug.  22. 
"    Sept.  19. 

"    Nov.  15. 
"    Nov.  16. 
"     Dec.  15. 

Hind. 
Hind. 
Pe  Gasparis. 

Goldschmidt. 
Hind. 
Hind 

2.2958 
2.4414 
2.4093 

2.4354 
2.9091 
2  6250 

3.479 
3.815 
3.740 

3.081 
4.962 
4263 

0.21723 
0.15792 
0.14383 

0.16204 
0.10361 
0  23180 

24. 
25 

Themis  .... 
Phocea  .... 

1853.  April  5. 
"    April  6. 

De  Gasparis. 
Chacornac 

3.1420 
24023 

5.570 
3723 

0.11701 
0  25335 

26. 

27. 
28 

Proserpine.. 
Euterpe  .... 
Bellona  .  .  . 

"    May  5. 

"    Nov.  8. 
1854,  March  1. 

Luther. 
Hind. 
Luther 

2.6556 
2.3473 

2  7784 

4.329 
3.596 
4  631 

0.08752 
0.17290 
0  15039 

29. 
30. 

31. 
32. 
33. 
34. 
35. 

36. 
37 

Ampliitrite.  . 
Urania  .... 

Euphrosyne 
Pomona.  .  .  . 
Polyhymnia. 
Circe  
Loucothea.  . 

Atlanta  
Fides  

"    March  1. 
"    July  22. 

"    Sept.  1. 
"    Oct.  26. 
"    Oct.  28. 
1855,  April  6. 
"     April  19. 

"     Oct.  5. 
"     Oct  5. 

Marth. 
Hind. 

Ferguson. 
Goldschmidt. 
Chacornac. 
Chacornac. 
Luther. 

Goldschmidt. 
Luther 

2.5548 
2.  3  642 

3.1561 
2.5831 
2.8646 
2.6839 
3.0060 

2.7487 
2  6422 

4.084 
3.635 

5.607 
4.160 
4.848 
4.397 
5.215 

4.557 
4  295 

0.07238 
0.12718 

0.21601 
0.08240 
0.33769 
0.10961 
0.21363 

0.29788 
0  17489 

38 

Leda 

1856  Jan   12 

Chacornac 

2  7399 

4  535 

0  15552 

39. 

40. 

41. 
4-> 

Lsetitia  
Harmouia  .  . 

Daphne  

"     Feb.  8. 
"     March  31. 

"    May  22. 
"     May  23. 

Chacornac. 
Luther. 

Goldschmidt. 
Pogson 

2.7710 
2.2679 

2.7674 
2  4401 

4.613 
3.415 

4.605 
3  812 

0.11081 
0.04608 

0.27034 
0  22566 

43. 
44. 
45. 

Ariadne  .... 
Nysa  
Eugenia  .  .  . 

1857,  April  15. 
"     May  27. 
"    June  28. 

Pogson. 
Goldschmidt. 
Goldschmidt. 

2.2038 
2.4242 
2.7159 

3.272 
3.774 
4.476 

6.16756 
0.14933 
0.08200 

TABLE  II  (a)— CONTINUED. 


No. 

Name. 

Date  of  Discovery. 

Discoverer. 

Mean 
Distance. 

Sidereal 
Period. 
Yrs. 

Eccentricity. 

46. 

Hestia  

1857,  Aug.  16. 

Pogson. 

2.5178 

3.995 

0  16184 

47 

Melete 

"    Sept     9 

G  oldsch  midt 

2  fi976 

4  189 

0  23686 

48. 

Aglnia  

"     Sept.  15. 

Luther. 

2  8831 

4896 

0  12788 

49 

Doris  

"     Sept.  19. 

Goldschmidt. 

3.1044 

5.470 

0  07580 

50. 

Pales  .  . 

"    Sept  19. 

Goldschmidt. 

3  0861 

5421 

0  23783 

61. 

52. 
53. 
54. 
65. 

56. 

57. 
58. 
59 

Virginia  .... 
Nemausa..  . 
Europa  .... 
Calypso  
Alexandra  .  . 

Pandora  .  .  . 
Mnemosyne 
Coucordia.  .  . 
Danae 

"    Oct.  4. 
1858,  Jan.  22. 
"    Feb.  6. 
"    April  4. 
"    Sept.  10. 

"    Sept.  10. 
1859,  Sept.  22. 
1860,  March  24. 
"'    Sept     9 

Ferguson. 
Laurent. 
Goldsohmidt. 
Luther. 
Goldschmidt. 

Searle. 
Luther. 
Luther. 
Goldschmidt 

2.6486 
23779 
3.0999 
2.6102 
2.7076 

2.7692 
3.1597 
2.6979 
2  9746 

4310 

3.667 
5.458 
4217 
4.553 

4.608 
5.616 
4.431 
5  131 

0.28695 
0.06285 
0.00450 
0.21263 
0.19941 

0.13895 
0.10752 
0.04103 
0  16308 

60. 
61  . 

Olympia  .«.  . 
Erato  

"    Sept.  12. 
"    Sept.  14. 

Chacornac. 
Forster. 

2.7147 
3.1296 

4.472 
5.537 

0.11883 
0  16964 

6?, 

Echo  

"    Sept   14. 

Ferguson 

2.3939 

3  729 

0  18543 

63. 
64. 
65. 

Ausonia  .... 
Angelina.... 
Cybele  

1861,  Feb.  10. 
"     March  4. 
f<    March  8. 

De  Gasparis 
Tempel. 
Tern  pel. 

2.3972 
2.6783 
3.4205 

3.712 

4.385 
6658 

0.12732 
0.12482 
0  12030 

66. 

Maia 

"    April    9. 

H  P  Tuttle 

2  6539 

4322 

0  15422 

67 

Asia  

"    April  17. 

Posrson 

2.4209 

3769 

0  18443 

68. 
69. 

Hesperia  .  .  . 
Leto  

"    April  29. 
"     April  29 

Schiaparelli. 
Luther 

2.9949 

27748 

5.186 
4622 

0.17452 
0  18566 

70. 

71. 

7? 

Panopea  .  .  . 

Feronia  .... 
Niobe  ..... 

"    May  5. 

"     May  29. 
"    Aug.  13. 

Goldschmidt. 

C.  H.  F.  Peters 
Luther. 

26132 

2.2660 
2.7555 

4.224 

3.411 

4.574 

0.18309 

0.11977 
0  17374 

73. 

Clytie  . 

1862  April  7 

Tuttle 

2  6648 

4  350 

0  04424 

74. 
75. 

76 

77 

Galatea  .... 
Eurydice  ... 

Freia  
Frieda 

"     Aug.  30. 
"    Sept.  22. 

"     Oct.  21. 
"    Nov  15 

Temple. 
C.  H.  F.  Peters 

D'  Arrest. 
C  H  F  Peters 

2.7777 
2.6698 

3.3877 
2  6719 

4.629 
4.362 

6.235 
4  368 

0.23820 
0.30690 

0.18772 
0  13582 

78 

Diana  

1863   March  15. 

Luther. 

2  6228 

4248 

0  20549 

79. 

80. 

81. 

82. 
83. 
84 

Eurynome.  . 
Sappho  

Terpsichore  . 
Alemene  .  .  . 
Beatrix  
Clio  .  .  . 

"    Sept.  14. 
1864,  May  2. 

"     Sept  30. 
"    Nov.  27. 
1865.  April  26. 
"    Aug  25 

Watson. 
Pogson. 

Tempel. 
Luther. 
De  Gasparis. 
Luther 

2.4433 
2.2963 

2.8563 
2.7603 
2.4287 
2  3674 

3.819 
3.480 

4.827 
4.586 
3.785 
3  643 

0.19509 
0.20022 

0.21175 
0.22599 
0.08410 
0  23754 

85. 

lo  

"     Sept  19 

C.  H  F  Peters 

2  6594 

4337 

0  19395 

86. 

87. 

Semele  
Sylvia  

1866,  Jan.  4. 
"    May  17. 

Tietjen. 
Pogson. 

30908 

5.434 

0.20493 

88. 
89. 

Thisbe  

u    June  15. 
"     Aug.  6. 

C.  H.  F.  Peters 

2.7503 
2.5341 

4.561 
4.032 

0.16670 
0.20499 

TABLE  VIII. 


Mean  Astronomical  Refractions. 
Barometer  30  in.     Thermometer,  Fah.  50°. 


Ap.Alt 

Reft-. 

Ap.  Alt 

Refr. 

Ap.  Alt 

Reft. 

Alt. 

Refr. 

0°  0' 

33'  51" 

4°  0' 

11'  52" 

12°  0' 

4'  28.1" 

42° 

l'.4.6" 

5 

32  53 

10 

11  30 

10 

4  24.4 

43 

1  2.4 

10 

31  58 

20 

11  10 

20 

4  20.8 

44 

1  0.3 

15 

31  5 

30 

10  50 

30 

4  17.3 

45 

0.58.1 

20 

30  13 

40 

10  32 

40 

4  13.9 

46 

56.1 

25 

29  24 

50 

10  15 

50 

4  10.7 

47 

54.2 

30 

28  37 

5  0 

9  58 

13  0 

4  7.5 

48 

52.3 

35 

27  51 

10 

9  42 

10 

4  4.4 

49 

50.5 

40 

27  6 

20 

9  27 

20 

4  1.4 

50 

48.8 

45 

26  24 

30 

9  11 

30 

3  58.4 

51 

47.1 

50 

25  43 

40 

8  58 

40 

3  55.5 

52 

45.4 

55 

25  3 

50 

8  45 

50 

3  52.6 

53 

43.8 

1  0 

24  25 

6  0 

8  32 

14  0 

3  49.9 

54 

42.2 

5 

23  48 

10 

8  20 

10 

3  47.1 

55 

40.8 

10 

23  13 

20 

8  9 

20 

3  44.4 

56 

39.3 

15 

22  40 

30 

7  58 

30 

3  41.8 

57 

37.8 

20 

22  8 

40 

7  47 

40 

3  39.2 

58 

36.4 

25 

21  37 

50 

7  37 

50 

3  36.7 

59 

350 

30 

21  7 

7  0 

7  27 

15  0 

3  34.3 

60 

33.6 

35 

20  38 

10 

7  17 

15  30 

3  27.3 

61 

32.3 

40 

20  10 

20 

7  8 

16  0 

3  20.6 

62 

31.0 

45 

19  43 

30 

6  59 

16  30 

3  14.4 

63 

29.7 

50 

19  17 

40 

6  51 

17  0 

3  8.5 

64 

28.4 

55 

18  52 

50 

6  43 

17  30 

3  2.9 

65 

27.2 

2  0 

18  29 

8  0 

6  35 

18  0 

2  57.6 

66 

25.9 

5 

18  5 

10 

6  28 

19 

2  47.7 

67 

24.7 

10 

17  43 

20 

6  21 

20 

2  38.7 

68 

23.5 

15 

17  21 

30 

6  14 

21 

2  30.5 

69 

22.4 

20 

17  0 

40 

6  7 

22 

2  23.2 

70 

21.2 

25 

16  40 

50 

6  0 

23 

2  16.5 

71 

19.9 

30 

16  21 

9  0 

5  54 

24 

2  10.1 

72 

18.8 

35 

16  2 

10 

5  47 

25 

2  4.2 

73 

17.7 

40 

15  43 

20 

5  41 

26 

58.8 

74 

16.6 

45 

15  25 

30 

5  36 

27 

53.8 

75 

15.5 

50 

15  8 

40 

5  30 

28 

49.1 

76 

14.4 

55 

14  51 

50 

5  25 

29 

44.7 

77 

13.4 

3  0 

14  35 

10  0 

5  *0 

30 

40.5 

78 

12.3 

5 

14  19 

10 

5  15 

31 

36.6 

79 

11.2 

10 

14  4 

20 

5  in 

32 

33.0 

80 

10.2 

15 

13  50 

30 

5  5 

33 

29.5 

81 

9.2 

20 

13  35 

40 

5  (/ 

34 

26.1 

82 

8.2 

25 

13  21 

50 

4  56 

35 

23.0 

83 

7.1 

30 

13  7 

11  0 

4  51 

36 

20.0 

84 

6.1 

35 

12  53 

10 

4  47 

37 

17.1 

85 

5.1 

40 

12  41 

20 

4  43 

38 

14.4 

86 

4.1 

45 

12  28 

30 

4  39 

39 

11* 

87 

3.1 

50 

12  16 

40 

4  35 

40 

9.3 

88 

2.0 

55 

12  3  I 

50 

4  31 

41 

6.9 

89 

1.0 

TABLE  IX 


Corrections  of  Mean  Refractions. 


Ap.Alt. 

1  differ 
-HB 

dif.  for 
—  1°F 

Ap.Alt. 

Dif.  for 

-KB. 

Dif.  for 

-1°F 

Ap.  Alt. 

Dif.  fo 
-}-lR 

Dif.  for 
-1°F. 

AH. 

Dif.  for 

-HB. 

Dif.  for 
1°  F. 

O  ' 

0  0 

74 

8.1 

4  0 

24.1 

1.70 

12  0 

9.00 

0.556 

42 

2.16 

0.130 

5 

71 

7.6 

10 

23.4 

1.64 

10 

8.86 

.548 

43 

2.09 

.125 

10 

69 

7.3 

20 

22.7 

1.58 

20 

8.74 

.541 

44 

2.02 

.120 

15 

67 

7.0 

30 

22.0 

1.53 

30 

8.63 

.533 

45 

1.95 

.116 

20 

65 

6.7 

40 

21.3 

1.48 

40 

8.51 

.524 

46 

1.88 

.112 

25 

63 

6.4 

50 

20.7 

1.43 

50 

8.41 

.517 

47 

1.81 

.108 

30 

61 

6.1 

5  0 

20.1 

1.38 

13  0 

8.30 

.509 

48 

1.75 

.104 

35 

59 

5.9 

10 

19.6 

1.34 

10 

8.20 

.503 

49 

1.69 

.101 

40 

58 

5.6 

20 

19.1 

1.30 

20 

8.10 

.496 

50 

1.63 

.097 

45 

56 

5.4 

30 

18.6 

1.26 

30 

8.00 

.490 

51 

1.58 

.094 

50 

55 

5.1 

40 

18.1 

1.22 

40 

7.89 

.482 

52 

1.52 

.090 

55 

53 

4.9 

50 

17.6 

1.19 

50 

7.79 

.476 

53 

1.47 

.088 

1  0 

52 

4.7 

6  0 

17.2 

1.15 

14  0 

7.70 

.469 

54 

1.41 

.085 

5 

50 

4.6 

10 

16.8 

1.11 

10 

7.61 

.464 

55 

1.36 

.082 

10 

49 

4.5 

20 

16.4 

1.09 

20 

7.52 

.458 

56 

1.31 

.079 

15 

48 

4,4 

30 

16.0 

1.06 

30 

7.43 

.453 

57 

1.26 

.076 

20 

46 

4.2 

40 

15.7 

1.03 

40 

7.34 

.448 

58 

1.22 

.073 

25 

45 

4.0 

50 

15.3 

1.00 

50 

7.26 

.444 

59 

1.17 

.070 

30 

44 

3.9 

7  0 

150 

0.98 

15  0 

7.18 

.439 

60 

1.12 

.067 

35 

43 

3.8 

10 

14.6 

.95 

15  30 

6.95 

.424 

61 

1.08 

.065 

40 

42 

3.6 

20 

14.3 

.93 

16  0 

6.73 

.411 

62 

1.04 

.062 

45 

40 

3.5 

30 

14.1 

.91 

16  30 

6.51 

.399 

63 

.99 

.060 

50 

39 

3.4 

40 

13.8 

.89 

17  0 

6.31 

.386 

64 

.95 

.057 

55 

39 

3.3 

50 

13.5 

.87 

17  30 

6.12 

.374 

65 

.91 

.055 

2  0 

38 

3.2 

8  0 

13.3 

.85 

18  0 

5.94 

.362 

66 

.87 

.052 

5 

37 

3.1 

10 

13.1 

.83 

19 

5.61 

.340 

67 

.83 

.050 

10 

36 

3.0 

20 

12.8 

.82 

20 

5.31 

.322 

68 

.79 

.047 

15 

36 

2.9 

30 

12.6 

.80 

21 

5.04 

.305 

69 

.75 

.045 

20 

35 

2.8 

40 

12.3 

.79 

22 

4.79 

.290 

70 

.71 

.043 

25 

34 

2.8 

50 

12.1 

.77 

23 

4.57 

.276 

71 

.67 

.040 

30 

33 

2.7 

9  0 

11.9 

.76 

24 

4.35 

.204 

72 

.63 

.038 

35 

33 

2.7 

10 

11.7 

.74 

25 

4.16 

.252 

73 

.59 

.036 

40 

32 

2.6 

20 

11.5 

.73 

26 

3.97 

.241 

74 

.56 

033 

45 

32 

2.5 

30 

11.3 

.72 

27 

3.81 

.230 

75 

.52 

.031 

50 

31 

2.4 

40 

11.1 

.71 

28 

3.65 

.219 

76 

.48 

.029 

55 

30 

2.3 

50 

11.0 

.70 

29 

3.50 

.209 

77 

45 

.027 

3  0 

30 

2.3 

10  0 

10.8 

.69 

30 

3.36 

.201 

78 

.41 

.025 

5 

29 

2.2 

10 

10.6 

.67 

31 

3.23 

.193 

79 

.38 

.023 

10 

29 

2.2 

20 

10.4 

.65 

32 

3.11 

.186 

80 

.34 

.021 

15 

28 

2.1 

30 

10.2 

.64 

33 

2.99 

.179 

81 

.31 

.018 

20 

28 

2.1 

40 

10.1 

.63 

34 

2.88 

.173 

82 

.27 

.016 

25 

27 

2.0 

50 

9.9 

.62 

35 

2.78 

.167 

83 

.24 

.014 

30 

27 

2.0 

11  0 

9.8 

.60 

36 

2.68 

.161 

84 

.20 

.012 

35 

26 

2.0 

10 

9.6 

.59 

37 

2.58 

.155 

85 

.17 

.010 

40 

26 

1.9 

20 

9.5 

.58 

38 

2.49 

.149 

86 

.14 

.008 

45 

25 

1.9 

30 

9.4 

.57 

39 

2.40 

.144 

87 

.10 

.006 

50 

25 

1.9 

40 

9.2 

.56 

40 

2.32 

.139 

88 

.07 

.004 

55 

25 

1.8 

50 

9.1 

.55 

41 

2.24 

.134 

89 

.03 

.002 

TABLE  X. 


Parallax  of  the  Sun,  on  the  first  day  of  each  Month:  the  mean 
horizontal  Parallax  being  assumed  =  8  ".60. 


Alti- 
tude. 

Jan. 

Feb. 
Dec. 

March.   April. 
NOT.       Oct. 

May. 
Sept. 

June. 
Aug. 

July. 

0 

// 

// 

., 

••       i    «       j    // 

.// 

0 

8.75 

8.73 

8.67 

8.60 

8.53 

8.48 

8.46 

5 

8.73 

8.69 

8.64 

8.56 

8.50 

8.44 

8.42 

10 

8.62 

8.59 

8.54 

8.47 

8.40 

8.35 

8.33 

15 

8.45 

8.43 

8.38 

8.30 

8.24 

8.19 

8.17 

20 

8.22 

8.20 

8.15 

8.08 

8.01 

7.97 

7.95 

25 

7.93 

7.91 

7.86 

7.79 

7.73 

7.68 

7.67 

30 

7.58 

7.56 

7.51 

7.45 

7.39 

7.34 

7.33 

35 

7.17 

7.15 

7.11 

7.04 

6.99 

6.94 

6.93 

40 

6.70 

6.68 

6.64 

6.59 

6.53 

6.49 

6.48 

4£ 

6.19 

6.17 

6.13 

6.08 

6.03 

5.99 

5.98 

50 

5.62 

5.61 

5.58 

5.53 

5.48 

545 

5.44 

55 

5.02 

5.01 

4.98 

4.93 

4.89 

4.86 

4.85 

60 

4.37 

4.36 

4.34 

4.30 

4.26 

4.24 

4.23 

65 

3.70 

3.69 

3.67 

3.63 

3.60 

358 

3.57 

70 

2.99 

2.98 

2.97 

2.94 

2.92 

2.90 

2.89 

75 

2.26 

2.26 

2.25 

2.23 

2.21 

2.19 

2.19 

80 

1.52 

1.52 

1.51 

1.49 

1.48 

1.47 

1.47 

85 

0.76 

0.76 

0.76 

0.75 

0.74 

0.74 

0.74 

90 

0.00 

0.00 

0.00 

0.00 

000 

0.00 

0.00 

TABLE  XL 
Semi-diurnal  Arcs. 


T      A. 

Declination. 

Lat. 

1° 

5° 

10° 

15° 

20° 

25° 

30o 

o 

k  m 

h  m 

h.  m 

h  m 

h  m 

h  m 

h  m 

5 

6  0 

6  2 

6  4 

6  5 

6  7 

6  9 

6  12 

10 

6  1 

6  4 

6  7 

6  11 

6  15 

6  19 

6  24 

15 

6  1 

6  5 

6  11 

6  16 

6  22 

6  29 

6  36 

20 

6  1 

6  7 

6  15 

6  22 

6  30 

6  39 

6  49 

25 

6  2 

6  9 

6  19 

6  29 

6  39 

6  50 

7  2 

30 

6  2 

6  12 

6  23 

6  36 

6  49 

7  2 

7  18 

35 

6  3 

6  14 

6  28 

6  43 

6  59 

7  16 

7  35 

40 

6  3 

6  17 

6  34 

6  52 

7  11 

7  32 

7  56 

45 

6  4 

6  20 

6  41 

7  2 

7  25 

7  51 

8  21 

50 

6  5 

6  24 

6  49 

7  14 

7  43 

8  15 

8  54 

55 

6  6 

6  29 

6  58 

7  30 

8  5 

8  47 

9  4* 

(SO 

6  7 

6  35 

7  11 

7  51 

8  36 

9  35 

IS  0 

65 

6  9 

6  43 

7  29 

8  20 

9  25 

12  0 

10 


TABLE   XII. 


Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time 
Argument,  Mean  Longitude  of  the  Sun. 


0* 

I* 

II* 

III* 

IV« 

V* 

0 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

0 

+  6  58.4 

—  1  29.7 

—  338.7 

+  1  27.0 

+  6    4.1 

+  2  49.7 

1 

639.7 

142.0 

334.2 

140.1 

6    6.3 

2  34.5 

2 

6  20.9 

1  53.8 

329.1 

1  53.1 

6    8.0 

2  18.9 

3 

6    2.1 

2    5.2 

323.5 

2    6.0 

6    9.1 

2    2.8 

4 

543.3 

2  15.9 

317.3 

2  18.9 

6    9.5 

146.4 

5 

524.5 

226.1 

3  10.7 

231.7 

6    9.3 

1  29.5 

6 

5    5.7 

235.9 

3    3.5 

244.3 

6    8.5 

1  12.3 

7 

446.9 

245.0 

2  56.0 

256.7 

6    7.2 

054.6 

8 

428.2 

253.6 

247.9 

3    8.9 

6    5.2 

036.6 

9 

4    9.6 

3    1.8 

239.5 

320.8 

6    2.5 

+  018.2 

10 

351.1 

3.   9.3 

230.5 

332.5 

559.3 

—  0    0.4 

11 

332.6 

3  16.3 

2  21.2 

343.9 

555.4 

019.5 

12 

3  14.3 

322.8 

2  11.5 

355.0 

551.0 

038.8 

13 

256.2 

328.6 

2    1.4 

4    5.8 

545.8 

058.4 

14 

238.3 

333.9 

1  51.0 

4  16.3 

540.1 

1  18.2 

15 

220.5 

338.6 

140.1 

426.5 

533.7 

1  38.3 

16 

2    3.0 

342.7 

1  29.0 

436.3 

526.7 

1  58.5 

17 

145.7 

346.3 

1  17.6 

445.7 

5  19.2 

2  19.1 

18 

1  28.6 

349.2 

1    5.9 

454.7 

5  11.1 

239.8 

19 

1  11.7 

351.5 

054.1 

5    3.3 

5    2.3 

3    0.7 

20 

055.2 

353.3 

042.0 

511.3 

453.0 

321.6 

21 

039.1 

354.4 

029.6 

5  18.9 

443.1 

342.8 

22 

023.3 

355.0 

0  17.1 

526.0 

432.7 

4    4.0 

23 

+  0    7.8 

3  55.0 

—  0    4.4 

532.6 

421.6 

425.3 

24 

—  0    7.3 

3  54.5 

+  0    8.4 

538.6 

4  10.1 

446.6 

25 

022.0 

353.3 

021.5 

544.2 

357.9 

5    8.1 

26 

036.3 

351.5 

034.5 

5  49.3 

345.3 

529.5 

27 

050.3 

349.2 

047.6 

553.9 

332.1 

551.0 

28 

1    3.8 

346.2 

1    0.7 

557.8 

3  18.5 

6  12.3 

29 

1  16.9 

342.8 

1  13.8 

6    1.2 

3    4.3 

633.7 

30 

—  1  29.7 

—  338.7 

+  1  27.0 

+  6    4.1 

-f  2  49.7 

—  654.9 

TABLE  XIII. 

Secular   Variation  of  Equation  of  Time, 
Argument,  Sun's  Mean  Longitude. 


0* 

I* 

II* 

III* 

IV*  |  V* 

sec. 

sec. 

sec. 

sec. 

sec. 

sec. 

BBC. 

0 

—  3 

+  4 

+  11 

+  14 

f  13 

-f  9 

3 

2 

5 

11 

14 

13 

8 

6 

1 

6 

12 

14 

12 

8 

9 

—  1 

6 

12 

15 

12 

7 

12 

0 

7 

12 

14 

12 

Ml 
* 

15 

+  1 

8 

13 

14 

11 

6 

18 

2 

8 

13 

14 

11 

6 

21 

2 

9 

14 

14 

10 

5 

24 

3 

9 

14 

14 

10 

5 

27 

4 

10 

14 

14 

9 

4 

30 

+  4 

+  11 

-f  14 

-r  13 

+  9 

+  4 

TABLE   XII 


11 


Equation  of  Time,  to  convert  Apparent  Time  into  Mean  Time. 
Argument,  Mean  Longitude  of  the  Sun. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

o 

mm.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

min.  sec. 

0 

—   654.9 

—  15  18.9 

—  13  58.7 

—   1  30.6 

+  11  30.0 

+  14    3.1 

1 

7  16.1 

15  27.9 

13  43.0 

1    0.2 

11  47.0 

13  56.0 

2 

737.2 

1536.1 

13  26.3 

—  029.8 

12    3.3 

1348.4 

3 

758.3 

1543.7 

13    8.9 

+    0    0.6 

12  18.7 

1340.1    j 

4 

8  19.1 

15  50.5 

12  50.5 

031.0 

12  33.4 

1331.1 

5 

839.8 

15  56.5 

1231.4 

1    1.3 

12  47.2 

1321.6 

6 

9    0.2 

16    1.8 

12  11.6 

1  31.4 

13    0.1 

1311.4 

7 

920.5 

16    6.3 

1151.1 

2    1.3 

13  12.2 

13    0.7 

8 

940.6 

16    9.9 

11  29.9 

231.0 

13  23.5 

12  49.4 

9 

10    0.3 

16  12.9 

11    7.9 

3    0.5 

13  33.9 

12  37.4 

10 

10  19.8 

16  15.1 

10  45.4 

329.7 

1343.6 

12  25.0 

11 

10  38.9 

16  16.5 

10  22.0 

358.6 

13  52.3 

12  12.2 

12 

10  57.8 

16  17.0 

958.1 

427.1 

14    0.2 

11  58.9 

13 

11  16.2 

16  16.6 

933.5 

455.2 

14    7.3 

11  45.1    '. 

14 

11  34.4 

16  15.4 

9    8.4 

522.9 

14  13.5 

11  30.9 

15 

11  52.1 

16  13.4 

842.6 

550.2 

14  18.9 

11  16.3 

16 

12    9.5 

16  10.4 

8  16.4 

6  17.1 

14  23.4 

11    1.1 

17 

12  26.5 

16    6.7 

749.6 

643.5 

14  27.2 

10  45.6 

18 

12  42.9 

16    2.1 

722.5 

7    9.3 

14  30.0 

10  29.7 

19 

12  58.9 

15  56.6 

6  54.9 

734.6 

14  32.  1 

10  13.5 

20 

13  14.4 

1550.1 

627.0 

759.3 

14  33.3 

956.9 

21 

13  29.5 

15  42.9 

558.5 

823.4 

14  33.7 

940.1 

22 

1344.1 

15  34.8 

529.7 

846.9 

14  33.3 

923.0 

23 

13  58.0 

15  25.8 

5    0.5 

9    9.8 

14  32.2 

9    5.7 

24 

1411.4 

15  16.0 

431.0 

932.0 

14  30.2 

848.0 

25 

1424.1 

15    5.2 

4    1.4 

953.5 

14  27.5 

830.2 

26 

14  36.3 

14  53.6 

331.6 

10  14.3 

14  24.0 

8  12.2 

27 

14  47.9 

1441.1 

3    1.5 

10  34.4 

14  19.9 

754.0 

28 

14  58.8 

14  27.7 

231.3 

10  53.8 

14  15.0 

735.5 

29 

15    9.2 

14  13.6 

2    1.0 

11  12.3 

14    9.4 

717.0 

30 

—  15  18.9 

—  1358.7 

—    1  30.6 

-HI  30.0 

-|   14    3.1 

+    658.4 

TABLE  XIII. 

Secular   Variation  of  liquation  of  Time. 
Argument,  Sun's  Mean  Longitude. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI* 

o 

sec. 

sec. 

sec. 

tec. 

sec. 

sec. 

0 

+4 

—  2 

—10 

—15 

—15 

—10 

3 

3 

3 

10 

15 

14 

10 

6 

3 

4 

11 

15 

14 

9 

9 

2 

4 

12 

15 

14 

8 

12 

1 

5 

12 

15 

13 

8 

15 

-fl 

6 

13 

15 

13 

7 

18 

0 

7 

13 

15 

12 

6 

21 

0 

7 

14 

15 

12 

5 

24 

—1 

8 

14 

15 

11 

5 

27 

2 

9 

15 

15 

11 

4 

30 

—  2 

—10 

—15 

—15 

—10 

—  9 

.12 


TABLE  XIV. 


Perturbations  of  Equation  of  Time. 


III. 


n. 

0 

100 

200  300 

400 

500 

600 

700 

800 

900 

1000 

sec. 

sec. 

sec.  ,  sec. 

sec. 

sec. 

sec. 

sec. 

gee. 

gee. 

sec. 

0 

1.4 

0.8 

.0 

1.7 

1.7 

1.2 

0.7 

0.4 

0.6 

1.4 

1.4 

100 

1.2 

1.4 

.1 

1.0 

1.6 

1.8 

1.1 

0.7 

0.6 

0.7 

1.2 

200 

0.9 

1.0 

.2  1.2 

1.2 

1.5 

1.7 

1.1 

0.5 

0.7 

0.9 

300 

0.7 

1.1 

.1  0.9 

1.2 

1.4 

1.5 

1.6 

1.2 

0.5 

0.7 

400 

0.5 

0.6 

.2  1.2 

0.8 

1.0 

1.6 

1.7 

1.5 

1.2 

0.5 

500 

1.0 

0.5 

0.6  1.2 

1.4 

0.8 

0.8 

1.5 

1.9 

1.5 

1.0 

600 

1.7 

1.0 

0.4  !  0.5 

1.2 

1.4 

0.9 

0.6 

1.3 

2.0 

1.7 

700 

1.9 

1.8 

.1 

0.4 

04 

1.1 

1.6 

1.1 

0.7 

1.2 

1.9 

800 

1.2 

1.8 

.8 

1.2 

0.4 

0.3 

1.0 

1.6 

1.2 

0.7 

1.2 

900 

0.7 

1.1 

.7 

1.8 

1.2 

0.6 

0.2 

0.8 

1.6 

1.3 

0.7 

1000 

1.4 

0.8 

.0 

1.7 

1.7  1.2 

0.7 

0.4 

0.6 

1.4 

1.4 

II. 

IV. 

sec. 

sec. 

sec. 

sec. 

sec. 

sec.   sec. 

sec. 

sec. 

sec. 

sec. 

0 

0.6 

0.7 

0.5 

0.3 

0.2 

0.6 

0.7 

0.5 

0.2 

0.1 

0.6 

100 

0.2 

0.7 

0.6 

0.5 

0.2 

0.3 

0.6 

0.9 

0.5 

0.2 

0.2 

200 

0.2 

0.4 

0.6 

0.5 

0.4 

0.3 

0.4 

0.6 

0.5 

0.5 

0.2 

300 

0.4 

0.2 

0.5 

0.5 

0.5 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

400 

0.5 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.4 

0.4 

0.5 

500 

0.4 

0.5 

0.5 

0.5 

0.4 

0.4 

0.3 

0.4 

0.5 

0.3 

0.4 

600 

0.3 

0.3 

0.5 

0.6 

0.4 

0.4 

0.3 

0.5 

0.7 

0.4 

0.3 

700 

0.4 

0.2 

0.3 

0.6 

0.6 

0.4 

0.2 

0.2 

0.7 

0.7 

0.4 

800 

0.6 

0.3 

0.2 

0.3 

0.7 

06 

0.3 

02 

0.3 

0.8 

0.6 

900 

0.8 

0.5 

0.3 

0.1 

0.4 

0.7 

0.5 

0.3 

0.1 

0.5 

0.8 

1000 

0.6 

0.7 

0.5 

0.3 

0.2 

0.6 

0.7 

0.5  0.2 

0.1 

0.6 

II. 

V. 

0 

sec. 
1.0 

Sec. 

1.0 

sec. 
1.1 

sec. 
1.2 

sec. 

1.1 

sec. 
1.0 

sec. 
0.7 

sec. 
0.4 

sec. 
0.6 

sec. 
0.9 

sec. 
1.0 

100 

0.9 

0.9 

0.8  • 

1.0 

1.3 

1.3 

.0 

0.7 

0.4 

0.5 

0.9 

200 

0.5 

0.7 

0.7 

0.8 

1.0 

1.0 

.1 

1.2 

0.9 

0.3 

0.5 

300 

0.2 

0.5 

0.7 

0.7 

0.8 

1.2 

.5 

1.5 

.1 

0.5 

0.2 

400 

0.3 

0.2 

0.5 

0.7 

0.7 

0.9 

.3 

1.4 

.4 

.0 

0.3 

500 

0.8 

0.3 

0.2 

0.5 

0.7 

0.7 

.0 

1.4 

.4 

.4 

0.8 

600 

.3 

0.7 

0.3 

0.3 

0.5 

0.7 

0.9 

1.1 

.4 

.6 

.3 

700 

.5 

1.1 

0.7 

0.3 

0.4 

0.5 

0.8 

1.0 

.2 

4 

.5 

800 

.3 

1.3 

1.0 

0.7 

0.4 

0.4 

0.6 

0.8 

.0 

.2 

.3 

900 

.1 

1.2 

1.2 

1.0 

0.8 

0.6 

0.5 

0.6 

0.9 

.1 

.1 

1000 

.0 

1.0 

1.1 

1.2 

1.1 

1.0 

0.7 

0.4  0.6 

0.9 

.0 

Moon  and  Nutation. 

I. 

»ec. 
0.5 

sec.  1  sec.  I  sec. 
0.8  1  1.0!  1.0 

sec. 
08 

sec. 
0.5 

sec. 
0.2 

sec. 

0.0 

sec. 
0.0 

sec.  \  sec. 
0.2  0.5 

N. 

0.1 

0.1  |  0.2  0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

o.i  lo.i 

Constant  3«.0 


TABLE  XV. 


13 


For  converting  any  given  day  into  the  decimal  part  of  a  yeai 
of  365  days. 


Day 

Jan. 

Feb. 

March 

April 

May 

Juno 

1 

.000 

.085 

.162 

.247 

.329 

.414 

2 

.003 

.088 

.164 

.249 

.331 

.416 

3 

.006 

.090 

.167 

.252 

.334 

.419 

4 

.008 

.093 

.170 

.255 

.337 

.422 

5 

.011 

.096 

.173 

.258 

.340 

.425 

6 

.014 

099 

.175 

.260 

.342 

.427 

7 

.016 

.101 

.178 

.263 

.345 

.430 

8 

.019 

.104 

.181 

.266 

.348 

.433 

9 

.022 

.107 

.184 

.268 

.351 

.436 

10 

.025 

.110 

.186 

.271 

.353 

.438 

11 

.027 

.112 

.189 

274 

.356 

.441 

12 

.030 

.115 

.192 

.277 

.359 

.444 

13 

.033 

.118 

.195 

.279 

.362 

.446 

14 

.036 

.121 

.197 

.282 

.364 

.449 

15 

.038 

.123 

.200 

.285 

.367 

.452 

16 

.041 

.126 

.203 

.288 

.370 

.455 

17 

.044 

.129 

.205 

.290 

.373 

.458 

18 

•046 

.132 

.208 

.293 

.375 

.460 

19 

.049 

.134 

.211 

.296 

.378 

.463 

20 

.052 

.137 

.214 

.299 

.381 

.466 

21 

.055 

.140 

.216 

.301 

.384 

.468 

22 

.058 

.142 

.219 

.304 

.386 

.471 

23 

.060 

.145 

.222 

.307 

.389 

.474 

24 

.063 

.148 

.225 

.310 

.392 

.477 

25 

.066 

.151 

.227 

312 

.395 

.479 

26 

.068 

.153 

.230 

.315 

.397 

.4^2 

27 

.071 

.156 

.233 

.318 

.400 

.4S5 

28 

.074 

.159 

.236 

.321 

.403 

.488 

29 

.077 

.238 

.323 

.405 

.490 

30 

.079 

.241 

.326 

.408 

.493 

31 

.0*2 

.244 

.411 

TABLE  XV.,  Continued. 


For  converting  any  given  day  into  the  decimal  part  of  a.  yeai 
of  365  days. 


Day 

July 

August 

Sept. 

Oct 

Nov. 

Dec. 

1 

.496 

.581 

.666 

.748 

.833 

.915 

2 

.499 

.584 

.668 

.751 

.836 

.918 

3 

.501 

.586 

671 

.753 

.838 

.921 

4 

.504 

.589 

.674 

.756 

.841 

.923 

5 

.507 

.592 

.677 

.759 

.844 

.926 

6 

.510 

.595 

.679 

.762 

846 

.929 

7 

.512 

.597 

.682 

.764 

.849 

.931 

8 

.515 

.600 

.685 

.767 

.852 

.934 

9 

.518 

.603 

.688 

.770 

855 

.937 

10 

.521 

.605 

.690 

.773 

.858 

.940 

11 

.523 

.608 

.693 

.775 

.860 

.942 

12 

526 

611 

.696 

.778 

.863 

.945 

13 

.529 

614 

.699 

.781 

.866 

.948 

14 

.532 

.616 

.701 

.784 

.868 

.951 

15 

.534 

.619 

.704 

.786 

.871 

.953 

16 

.537 

.622 

.707 

.789 

.874 

.956 

17 

.540 

.625 

.710 

.792 

877 

.959 

18 

.542 

.627 

.712 

.795 

879 

.962 

19 

.545 

.630 

.715 

.797 

.882 

.964 

20 

.548 

.633 

.718 

.800 

885 

.967 

21 

.551 

.636 

.721 

.803 

888 

.970 

22 

.553 

.638 

.723 

805 

890 

.973 

23 

.556 

.641 

.726 

.808 

893 

.975 

24 

.559 

.644 

.729 

.811 

896 

.978 

25 

.562 

.647 

.731 

.814 

.899 

.981 

20 

564 

.649 

.734 

.816 

.901 

.984 

27 

.567 

.652 

.737 

.819 

.904 

.986 

28 

.570 

.655 

.740 

.822 

.907 

.989 

29 

.573 

.658 

.742 

.825 

.910 

.992 

30 

.575 

.660 

.745 

.827 

.912 

.995 

31 

.578 

.663 

.830 

.997 

TABLE  XVI. 


15 


For  converting  time  into  decimal  parts  of  a  day. 


Hours 

Minutes 

Seconds 

h. 

m.         m. 

8.              S. 

1 

.04167 

1 

.00069 

31 

.02153 

1 

.00001  -  31 

.00036 

2 

.08333 

2 

.00139 

32 

.02222 

2 

.00002   32 

.00037 

3 

.12500 

3 

.00208 

33 

.02292 

3 

.00003   33 

.00038 

4 

.16667 

4 

.00278 

34 

.02361 

4 

.00005  ;  34 

.00039 

5 

.20833 

5 

.00347 

35 

.02430 

5 

.00006  :  35 

.00040 

j 

6 

.25000 

6 

.00417 

36 

.02500 

6 

.00007   36 

.00042 

7 

.29167 

7 

.00486 

37 

.02569 

7 

.00008  f  37 

.00043 

8 

.33333 

8 

.00556 

38 

.02639 

8 

.00009   33 

.00044 

9 

.37500 

9   .00625 

39 

.02708 

9 

.00010  •  39 

.00045 

10 

.41667 

10 

.00694 

40 

.02778 

10 

.00012  .  40 

.00046 

11 

.45833 

11 

.00764 

41 

.02847 

11 

.00013  f  41 

.00047 

12 

.50000 

12 

.00833 

42 

.02917 

12 

.00014  ,  42 

.00049 

13 

.54167 

13  !  .00903 

43 

.02986 

13 

.00015 

43 

.00050 

14 

.58333 

14 

.00972 

44 

.03056 

14 

.00016 

44 

.00051 

15 

.62500 

15 

01042 

45 

.03125 

Ifi 

.00017 

45 

.00052 

16 

.66667 

16 

.01111 

46 

.03194 

16 

.00018 

46 

.00053 

17 

.70833 

17 

.01180 

47 

.03264 

17 

.00020 

47 

.00054 

18 

.75000 

18 

.01250 

48 

.03333 

18 

.00021 

48 

.00056 

19 

.79167 

19 

.01319 

49 

.03403 

19 

.00022 

49 

.00057 

20 

.83333 

20 

.01389 

50 

.03472 

20 

.00023 

50 

.00058 

21 

.87500 

21 

01458 

51 

.03542 

21 

.00024 

51 

.00059 

22 

.91667 

22 

.01528 

52 

.03611 

22 

.00025 

52 

.00060 

23 

.95833 

23 

01597 

53 

.03680 

23 

.00027 

53 

.00061 

24 

1.00000 

24 

.01667 

54 

.03750 

24 

.00028 

54 

.00062 

25 

.01736 

55 

.03819 

25 

.00029 

55 

.00064 

26 

.01805 

56 

.03889 

26 

.00030 

56 

.00065 

27 

.01875 

57 

.03958   27 

.00031 

57 

.00066 

28 

.01944 

58 

.04028   28 

.00032 

58 

.00067 

29 

.02014 

59 

.04097   29 

.00034 

59 

.00068 

30 

.02083 

60 

.04167   30 

.00035 

60 

.00069 

16 


TABLE  XVII. 


For  converting  Minutes  and  Seconds  of  a  degree,  into  the 
decimal  division  of  the  same. 


Minutes 

Seconds 

1 

.01667 

.  f  - 
31 

.51667 

1 

.00028 

31 

.00861 

2 

.03333 

32 

.53333 

2 

.00056 

32 

.00889 

3 

.05000 

33 

.55000 

3 

.00083 

33 

.00917 

4 

.06667 

34 

.56667 

4 

.00111 

34 

.00944 

5 

.08333 

35 

.58333 

5 

.00139 

35 

.00972 

6 

.10000 

36 

.60000 

6 

.00167 

36 

.01000 

7 

.11667 

37 

.61667 

7 

.00194 

37 

.01028 

8 

.13333 

38 

.63333 

8 

.00222 

38 

.01056 

9 

.15000 

39 

.65000 

9 

.00250 

39 

.01083 

10 

.16667 

40 

.66667 

10 

.00278 

40 

.01111 

11 

.18333 

41 

.68333 

11 

.00306 

41 

.01139 

12 

.20000 

42 

.70000 

12 

.00333 

42 

.01167 

13 

.21667 

43 

.71667 

13 

.00361 

43 

.01194 

14 

.23333 

44 

.73333 

14 

.00389 

44 

.01222 

15 

.25000 

45 

.75000 

15 

.00417 

45 

.01250 

18 

.26667 

46 

.76667 

16 

.00444 

46 

.01278 

17 

.28333 

47 

.78333 

17 

.00472 

47 

.01306 

18 

.30000 

48 

.80000 

18 

.00500 

48 

.01333 

19 

.31667 

49 

.81667 

19 

.00528 

49 

.01361 

20, 

.33333 

50 

.83333 

20 

.00556 

50 

.01389 

21 

.35000 

51 

.85000 

21 

.00583 

51 

.01417 

22 

.30667 

52 

.8tif>67 

22 

.00611 

52 

.01444 

23 

.38333 

53 

88333 

23 

.006:39 

53 

01472 

24 

.40000 

54 

.90000 

24 

.00667 

54 

01500 

25 

.41667 

55 

.91667 

25 

00694 

55 

.01528 

20 

.43333 

56 

.933:33 

26 

00722 

f)fi 

01556 

27 

.45000 

57 

.95000 

27 

00750 

57 

01583 

28 

.46667 

58 

.96667 

'28  !  00778 

58 

Olfill 

20 

.48333 

59 

98333   29   00806 

59   01639 

30  i  .50000 

60   1.00000  1  30  1  00833 

60   01667  ; 

TABLE  XYIII. 
Sun's  Epochs. 


17 


Years. 

M.  Long. 

Long.  Perl 

I. 

IL 

III. 

IV. 

V. 

N. 

VL 

VII. 

S  °   '   " 

,  •   '   " 

1830 

9  10  37  46.9 

9  10  0  54 

228 

279 

169 

598 

758 

519 

989 

362 

1831 

9  10  23  27.4 

9  10  1  55 

588 

278 

793 

130 

842 

573 

233 

396 

1832B. 

9  10  9  7.9 

9  10  2  57 

948 

278 

418 

661 

926 

627 

482 

430 

1833 

9  10  53  56.8 

9  10  3  59 

342 

280 

47 

194 

11 

681 

764 

464 

1834 

9  10  39  37.3 

9  10  5  0 

702 

279 

671 

725 

95 

734 

11 

498 

1835 

9  10  25  17.8 

9  10  6  2 

62 

279 

296 

256 

179 

788 

257 

532 

1836  B. 

9  10  10  58.4 

9  10  7  3 

422 

278 

920 

788 

264 

842 

504 

566 

1837 

9  10  55  47.2 

9  10  8  5 

816 

280 

549 

321 

348 

895 

787 

600 

1838 

9  10  41  27.8 

9  10  9  6 

176 

279 

173 

852 

432 

949 

33 

634 

1839 

9  10  27  8.3 

9  10  10  8 

536 

279 

798 

383 

517 

3 

279 

668 

1840  B. 

9  10  12  48.8 

9  10  11  9 

896 

278 

422 

915 

601 

56 

526 

702 

1841 

9  10  57  37.7 

9  10  12  11 

•290 

280 

51 

447 

685 

110 

809 

736 

1842 

9  10  43  18.2 

9  10  13  12 

650 

279 

676 

979 

770 

164 

55 

770 

1843 

9  10  28  58.8 

9  10  14  14 

10 

279 

300 

510 

854 

218 

301 

804 

1844  B. 

9  10  14  39.3 

9  10  15  15 

370 

278 

924 

41 

938 

272 

548 

838 

1845 

9  10  59  28.2 

9  10  16  17 

764 

280 

553 

574 

23 

325 

831 

872 

1846 

9  10  45  8.7 

9  10  17  19 

124 

280 

177 

106 

107 

379 

77 

906 

1847 

9  10  30  49.2 

9  10  18  20 

484 

279 

802 

637 

191 

433 

324 

940 

1848  B. 

9  10  16  29.8 

9  10  19  22 

844 

278 

427 

168 

276 

487 

570 

974 

1849 

9  11  1  18.6 

9  10  20  23 

238 

280 

55 

700 

360 

540 

853 

8 

1850 

9  10  46  59.2 

9  10  21  25 

598 

280 

680 

231 

444 

594 

99 

41 

1851 

9  10  32  39.7 

9  10  22  26 

958 

279 

304 

762 

529 

648 

346 

75 

1852  B. 

9  10  18  20.2 

9  10  23  28 

319 

278 

929 

294 

613 

701 

592 

109 

1853 

9  11  3  9.1 

9  10  24  29 

713 

280 

557 

827 

697 

755 

875 

143 

1854 

9  10  48  49.6 

9  10  25  31 

73 

280 

182 

358 

782 

809 

121 

177 

1855 

9  10  34  30.2 

9  10  26  32 

433 

279 

806 

889 

866 

863 

368 

211 

1856  B. 

9  10  20  10.7 

9  10  27  34 

793 

279 

430 

421 

950 

916 

614 

245 

1857 

9  11  4  59.6 

9  10  28  35 

187 

281 

60 

953 

35 

970 

897 

279 

1858 

9  10  50  40.1 

9  10  29  37 

547 

280 

684 

485 

119 

24 

144 

313 

1859 

9  10  36  20.7 

9  10  30  39 

907 

279 

308 

16 

203 

78 

390 

347 

1860  B. 

9  10  22  1.2 

9  10  31  40 

267 

279 

933 

547 

288 

131 

636 

381 

1861 

9  11  6  50.1 

9  10  32  42 

661 

281 

562 

80 

372 

185 

919 

415 

1862 

9  10  52  30.6 

9  10  33  43 

21 

280 

186 

612 

456 

239 

166 

449 

1863 

9  10  38  11.1 

9  10  34  45 

381 

280 

810 

143 

541 

292 

412 

483 

1864  B. 

9  10  23  51.7 

9  10  35  46 

741 

279 

435 

674 

625 

346 

659 

517 

1865 

9  11  8  40.5 

9  10  36  48 

135 

281 

64 

207 

709 

400 

941 

551 

1866 

9  10  64  21.1 

9  10  37  49 

495 

280 

688 

738 

794 

453 

188 

585 

1867 

9  10  40  1.6 

9  10  38  51 

855 

280 

313 

270 

878 

507 

434 

619 

1868  B. 

9  10  25  42.2 

9  10  39  52 

215 

279 

937 

801 

962 

561 

681 

653 

1869 

9  11  10  31.0 

9  10  40  54 

609 

281 

566 

334 

47 

615 

963 

687 

1870 

9  10  56  11.6 

9  10  41  56 

969 

280 

190 

865 

131 

668 

210 

721 

1871 

9  10  41  52.1 

9  10  42  57 

329 

279 

814 

396 

216 

742 

457 

755 

1872B. 

9  10  27  32.6 

9  10  43  59 

690 

278 

439 

928 

300 

775 

703 

789 

1873 

9  11  12  21.5 

9  10  45  0 

84 

280 

67 

461 

384 

829 

986 

823 

1874 

9  10  58  2.0 

9  10  46  2 

444 

280 

692 

992 

469 

883 

•232 

857 

1875 

9  10  43  42.6 

9  10  47  3 

804 

279 

316 

523 

553 

937 

479 

891 

1876  B. 

9  10  29  23.1 

9  10  48  5 

164 

279 

940 

55 

637 

990 

725 

925 

1877 

9  11  14  12.0 

9  10  49  6 

558 

281 

570 

587 

722 

44 

8 

959 

1878 

9  10  59  52.5 

9  10  50  8 

918 

280 

194 

119 

806 

98 

255 

993 

1879 

9  10  45  33.1 

9  10  51  10 

278 

279 

818 

650 

890 

152 

501 

27 

1880  B. 

9  10  31  13.6 

9  10  52  11 

638 

279 

443 

181 

975 

205 

747 

61 

1881 

9  11  16  2.5 

9  10  53  13 

32 

281 

72 

714 

59 

259 

30 

95 

1882 

9  11  1  43.0 

9  10  54  14 

392 

280 

696 

246 

143 

313 

277 

129 

1883 

9  10  47  23.5 

9  10  55  16 

752 

280 

320 

777 

228 

366 

523 

163 

1884  B. 

9  10  33  4.1 

9  10  56  17 

112 

279 

945 

308 

312 

420 

770 

197 

1 

18 


TABLE  XIX. 

Su?i*s  Motions  for  Months. 


Months 

M.   Long. 

Per. 

I 

II 

III 

IV 

V 

N 

VI 

VII 

«   0    '    " 

„ 

January 

0  0  0  0.0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

February 

1  0  33  18.2 

5 

47 

85 

138 

45 

7 

5 

125 

3 

,,  ,    (  Com. 

1  28  9  11.4 

10 

993 

.62 

263 

86 

14 

9 

141 

6 

March    \  Bis. 

1  29  8  19.8 

10 

27 

164 

267 

87 

14 

9 

178 

6 

.,    <  Com. 

2  28  42  29.7 

15 

42 

246 

401 

13] 

21 

13 

266 

8 

APnl    i  Bis. 

2  29  41  38.0 

15 

76 

249 

405 

132 

21 

13 

302 

8 

T..       (  Com. 

3  28  16  39.6 

20 

59 

329 

534 

175 

28 

18 

355 

11 

May   {  Bis. 

3  29  15  47.9 

20 

92 

331 

538 

176 

28 

18 

391 

11 

T       (  Com. 

4  28  49  57.9 

26 

110 

414 

672 

220 

35 

22 

480 

14 

June    \  Bis. 

4  29  49  6.2 

26 

144 

416 

676 

221 

35 

23 

516 

14 

,  ,      (  Com. 

5  28  24  7.8 

31 

129 

496 

806 

263 

41 

27 

569 

17 

J'lly   i  Bis. 

5  29  23  16.1 

31 

163 

499 

810 

265 

42 

27 

605 

17 

.       <  Com. 

6  28  57  26.1 

36 

182 

580 

943 

309 

49 

31 

694 

20 

AuS<    {  Bis. 

6  29  56  34.4 

36 

216 

583 

948 

310 

49 

31 

730 

20 

0       (  Com. 

7  29  30  44.2 

41 

233 

665 

81 

354 

56 

36 

819 

23 

SeP«     i  Bis. 

8  0  29  52.6 

41 

268 

668 

86 

355 

56 

36 

855 

23 

~  .     (  Com. 

8  29  4  54.1 

46 

250 

748 

215 

397 

63 

40 

908 

25 

Oct-   i  BIS. 

9  0  4  2.5 

46 

284 

750 

219 

399 

63 

40 

944 

25 

N      (  Com. 

9  29  38  12.5 

51 

300 

832 

353 

443 

70 

45 

33 

28 

Nov'    i  Bis. 

10  0  37  20.7 

51 

333 

835 

357 

444 

70 

45 

69 

28 

TV      5  Com. 

10  29  12  22.3 

56 

313 

915 

486 

486 

77 

49 

121 

31 

Dec-    \  Bis. 

11  0  11  30.6 

56 

347 

917 

491 

488 

77 

49 

158 

31 

TABLE  XX. 
Surfs  Motions  for  Days  and  Hours. 


Daysj 

M.  Long. 

>er. 

I 

II 

III 

IV 

V 

N 

VI 

VII 

Hrs. 

Long. 

I 
VI 

II 

III 

0   '    " 

// 

~,  7, 

1 

0  0  0.0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

2  27.8 

1 

0 

0 

2 

0  59  8.3 

0 

34 

3 

4 

1 

0 

0 

36 

0 

2 

4  55.7 

3 

0 

0 

3 

1  58  16.7 

0 

68 

5 

9 

3 

0 

0 

73 

0 

3 

7  23.5 

4 

0 

1 

4 

2  5725.0 

0 

101 

8 

13 

4 

1 

0 

109 

0 

4 

9  51.4 

6 

0 

5 

3  5633.3 

1 

135 

11 

18 

6 

1 

1 

145 

0 

5 

12  19.2 

7 

1 

6 

4  5541.6 

1 

169 

14 

22 

7 

1 

1 

181 

0 

6 

14  47.1 

8 

1 

7 

5  5450.0 

1 

203 

16 

27 

9 

1 

1 

218 

1 

7 

17  14.9 

10 

1 

8 

6  5358.3 

1 

236 

19 

31 

10 

2 

1 

254 

1 

8 

19  42.8 

11 

1 

9 

7  53  6.6 

1 

270 

22 

36 

12 

2 

1 

290 

1 

9 

22  10.6 

13 

1 

2 

10 

8  52  15.0 

1 

304 

25 

40 

13 

2 

1 

327 

1 

10 

24  38.5 

14 

1 

2 

11 

9  51  23.3 

2 

338 

27 

44 

15 

2 

1 

363 

1 

11 

27  6.3 

16 

1 

2 

12 

10  5031.6 

2 

371 

30 

49 

16 

2 

2 

399 

1 

12 

29  34.2 

17 

1 

2 

13 

11  4940.0 

2 

405 

33 

53 

17 

3 

2 

435 

1 

13 

32  2.0 

18 

1 

2 

14 

12  48  48.3 

2 

439 

36 

58 

19 

3 

2 

472 

1 

14 

34  29.9 

20 

2 

3 

15 

13  47  56.6 

2 

473 

38 

.62 

20 

3 

2 

508 

2 

15 

36  57.7 

21 

2 

3 

16 

14  47  4.9 

2 

506 

41 

67 

22 

3 

2 

544 

2 

16 

39  25.6 

23 

2 

3 

17 

15  46  13.3 

3 

540 

44 

71 

23 

4 

2 

581 

2 

17 

41  53.4 

24 

2 

3 

18 

16  4521.6 

3 

574 

47 

76 

25 

4 

2 

617 

2 

18 

44  21  2 

25 

2 

3 

19 

17  4429.9 

3 

608 

49 

80 

26 

4 

3 

653 

2 

19 

46  49.1 

27 

2 

4 

20 

18  4338.3 

3 

641 

52 

85 

28 

4 

3 

690 

2 

20 

49  16.9 

28 

2 

4 

21 

19  4246.6 

3 

675 

55 

89 

29 

5 

3 

726 

2 

21 

51  44.8 

30 

2 

4 

22 

20  41  54.9 

4 

709 

58 

93 

31 

5 

3 

762 

2 

22 

54  12.6 

31 

2 

4 

23 

21  41  3.3 

4 

743 

60 

98 

32 

5 

3 

798 

2 

23 

56  40.5 

32 

3 

4 

24 

22  40  11.6 

4 

777 

63 

102 

33 

5 

3 

835 

2 

24 

59  8.3 

34 

3 

4 

25 

23  39  19.9 

4 

810 

66 

107 

35 

5 

4 

871 

2 

£C 

24  38  28.2 

4 

844 

68 

111 

36 

6 

4 

907 

2 

27 

25  3736.6 

4 

878 

71 

116 

38 

6 

4 

943 

2 

28 

26  3644.9 

5 

912 

74 

120 

39 

6 

4 

980 

2 

29 

27  3553.2 

5 

945 

77 

125 

41 

6 

4 

16 

3 

30 

28  35  1.6 

5 

979 

79 

129 

42 

7 

4 

52 

3 

31 

29  34  9.9 

5 

13 

82 

134 

44 

7 

4 

89 

3 

TABLE  XXI. 

Sun's  Motions  for  Minutes  and  Seconds. 


TABLE  XXII.  19 

Mean  Obliquity  of 
the  Ecliptic. 


Min. 

Long. 

Min. 

Long. 

Sec. 

Lon. 

Sec.  Lon. 

,  „ 

/  // 

"TT~ 

tf 

1 

0  2.5 

31 

1  16.4 

1 

0.0 

31 

1.3 

2 

4.9 

32 

1  18.8 

2 

0.1 

32 

1.3 

3 

7.4 

33 

1  21.3 

3  0.1 

33 

.4 

4 

9.9 

34 

1  23.8 

4  0.2 

34 

.4 

5 

12.3 

35 

1  26.2 

5 

0.2  i  35 

.4 

6 

14.8 

36 

1  28.7 

6 

0.2  1  36 

.5 

7 

17.2 

37 

1  31.2 

7  0.3 

37 

.5 

8 

19.7 

38 

1  33.6 

8  0.3 

1  38 

.6 

9 

22.2 

39 

1  36.1 

9  i  0.4 

39 

.6 

10 

24.6 

40 

1  38.6 

10 

0.4 

40 

.6 

11 

27.1 

41 

1  41.0 

11 

0.5 

41 

.7 

12 

29.6 

42 

1  43.5 

12  0.5 

'  42 

.7 

13 

32.0 

43 

1  46.0 

13 

0.5 

43 

.8 

14 

34.5 

44 

1  48.4 

14 

0.6 

44 

.8 

15 

37.0 

45 

1  50.9 

15 

0.6 

45 

.8 

16 

39.4 

46 

1  53.3 

16 

0.7 

46 

.9 

17 

41.9 

47 

1  55.8 

17 

0.7 

47 

.9 

18 

44.4 

48 

1  58.3 

18 

0.7 

48 

2.0 

19 

46.8 

49 

2  0.7 

19 

0.8 

49 

2.0 

20 

49.3 

50 

2  3.2 

20 

0.8 

50 

2.0 

21 

51.7 

51 

2  5.7 

21 

0.9 

51 

2.1 

22 

54.2 

52 

2  8.1 

22 

0.9 

52 

2.1 

23 

56.7 

53 

2  10.6 

23 

0.9 

53 

2.2 

24 

59.1 

54 

2  13.1 

24 

.0 

54 

2.2 

25 

1  1.6 

55 

2  15.5 

25 

.0 

55 

2.3 

26 

1  4.1 

56 

2  18.0 

26 

.1 

56 

2.3 

27 

1  6.5 

57 

2  20.5 

27 

.1 

57 

2.3 

28 

1  9.0 

58 

2  22.9 

28 

.1 

58 

2.4 

29 

1  11.5 

59 

2  25.4 

29 

1.2 

59 

2.4 

30 

1  13.9 

60 

2  27.8 

30 

1.2 

60 

2.5 

Years 

23  27 

1835 

3880 

1836 

3835 

1837 

37.89 

1838 

37.43 

1839 

36.98 

1840 

36.52 

1841 

36.06 

1842 

35.61 

1843 

35.15 

1844 

34.69 

1845 

34.23 

1846 

33.78 

1847 

33.32 

1848 

32.86 

1849 

32.41 

1850 

31.95 

1851 

31.49 

1852 

31.04 

1853 

30.58 

1854 

30.12 

1855 

29.66 

1856 

29.21 

1857 

2875 

1858 

28'29 

1859 

27>4 

1860 

27.38 

1861 

2692 

1862 

26.47 

1863 

26.01 

1864 

25.55 

TABLE  XXIII. 

Sun's  Hourly  Motion. 

Argument.     Sun's  Mean  Anomaly. 


0* 

I* 

Ilf 

III* 

iv« 

V* 

0 

0 

10 
20 
30 

2  32.92 
2  32.84 
2  32.59 
2  32.20 

2  32.20 
2  31.67 
2  31.02 
2  30.28 

2  30.2S 
2  29.46 
2  28.61 
2  27.74 

2  27.74 
2  26.89 
2  26.07 
2  25.32 

2  25.32 
2  24.64 
2  24.06 
2  23.60 

2  23.60 
2  23.26 
2  23.05 
2  22.99 

o 
30 
20 
10 
0 

XI* 

X* 

IX* 

VIII* 

VII* 

VI- 

TABLE  XXIV. 
Sun's  Semi-diameter. 
Argument.    Sun's  Mean  Anomaly. 


0 

I* 

II* 

III« 

IV« 

V* 

o 
0 
10 
20 
30 

16  17.3 
16  17.0 
16  16.2 
16  15.0 

16  15.0 
16  13.3 
16  11.2 
16  8.8 

16  8.8 
16  6.2 
16  3.4 
16  0.6 

16  0.6 

15  57.8 
15  55.1 
15  52.7 

15  52.7 
15  50.5 
15  48.6 
15  47.0 

15  47.0 
15  45.9 
15  45.2 
15  45.0 

0 

30 
20 
10 
0 

XL 

X* 

IX* 

VIII* 

vii- 

Vis 

20 


TABLE  XXV. 

Equation  of  the  Surfs  Centre. 
Argument.     Sun's  Mean  Anomaly. 


0* 

If 

II* 

III* 

IV* 

V* 

0 

,       0        '        " 

O      '      " 

O       '      " 

O       '        " 

/    // 

0      '      " 

0 

11  29  59  13.9 

0  57  58.5 

1  40  10.7 

1  54  34.1 

38    4.8 

05552.6 

1 

0      0    1  1V.3 

0  59  43.9 

1  41     8.9 

1  54  30.5 

37    2.4 

0  54   8.7 

2 

0    3  20.6 

1     1  28.0 

1  42     5.1 

1  54  24.8 

35  58.1 

0  52  24.0 

? 

0    5  23  9 

1     3  10.9 

1  42  59.3 

1  54  17.0 

34  52.2 

0  5038.2 

4 

0    7  27.0 

1     4  52.6 

1  43  51.8 

1  54     7.1 

33  44.6 

0  4851.6 

5 

0    9  30.0 

1     6  33.0 

1  44  42.1 

1  53  55.2 

32  35.4 

0  47   4.2 

6 

0  11  32.8 

1     8  12.3 

1  45  30.4 

1  53  41.0 

31  24.4 

0  45  16.0 

7 

0  13  35.4 

1     9  50.1 

1  46  16.8 

1  53  24.9 

30  11.9 

04326.9 

8 

0  15  37.7 

1   11  26.5 

1  47     1.2 

1  53     6.7 

28  57.7 

0  41  37.0 

9 

0-17  39.6 

1   13    1.7 

1  47  43.5 

1  52  46.5 

27  42.0 

0  3946.5 

10 

0  19  41.2 

1   1435.3 

1  48  23.9 

1  52  24.2 

20  24.8 

03755.3 

11 

0  21  42.4 

1   16    7.5 

1  49     2.2 

1  51  59.8 

25    5.9 

0  36    3.3 

12 

0  23  43.1 

1   1738.2 

1  49  38.4 

1  51  33.4 

23  45.7 

0  34  10.8 

13 

0  25  43.4 

1   19    7.5 

1  50  12.6 

1  51     5.0 

22  23.8 

0  32  17.7 

14 

0  27  43.2 

1  2035.2 

1  30  44.7 

1  50  34.5 

21    0.6 

0  3023.8 

15 

0  29  42.3 

1  22    1.5 

1  51  14.9 

1  50     2.2 

1936.0 

02829.6 

16 

0  31  40.9 

1  23  26.0 

1  51  42.9 

1  49  27.7 

18    9.9 

0  26  34.8 

17 

0  33  38.9 

1  2443.9 

1  52    8.7 

1  48  51.3 

16  42.4 

0  2439.6 

18 

0  35  36.2 

1  26  10.3 

1  52  32.5 

1  48  13.0 

15  13.7 

0  2243.9 

19 

0  37  32.9 

1  2730.0 

1  52  54.3 

1  47  32.7 

13  43.5 

0  2047.9 

20 

0  39  28.8 

1  2848.0 

1  53  13.9 

1  46  50.4 

12  12.1 

0  1851.4 

21 

0  41  23.9 

1  30    4.2 

1  53  31.4 

1  46     6.3 

10  39.3 

0  1654.6 

22 

0  43  18.1 

1  31  18.8 

1  53  46.8 

1  45  20.3 

9    5.4 

0  1457.5 

23 

045  11.5 

1  3231.7 

1  54    0.1 

1  44  32.2 

730.3 

0  13    0.1 

24 

0  47    4.0 

1  3342.7 

1  54  11.2 

1  43  42.4 

5  54.0 

0  11    2.6 

25 

0  48  55.6 

1  3452.0 

1  54  20.4 

1  42  50.7 

1     4  16.5 

0    9    4.8 

26 

0  50  46.3 

1  35  59.4 

1  54  27.2 

1  41  57.1 

1     2  37.8 

0    7   6.9 

27 

0  52  36.0 

1  37    5.1 

1  54  32.1 

1  41     1.7 

1     0  58.0 

0    5    87 

28 

0  54  24.6 

1  38    8.8 

1  54  34.9 

1  40     4.5 

0  59  17.3 

0    3105 

29 

0  56  12.1 

1   39  10.8 

1  54  35.4 

1  39     5.6 

0  57  35.4 

0    1  12.2 

30 

0  57  58.5 

1  40  10.7 

1  54  34.1 

1  38     4.8 

0  55  52.6 

TABLE  XXVI. 

Secular   Variation  of  Equation  of  Surfs  Centre. 
Argument.    Sun's  Mean  Anomaly. 


0* 

I* 

II* 

m» 

IV* 

V* 

O 

0 

—  0 

—  9 

—  15 

—  17 

—  15 

—  8 

2 

1 

9 

15 

17 

14 

8 

4 

1 

10 

16 

17 

14 

7 

6 

2 

10 

16 

17 

14 

7 

8 

2 

11 

16 

17 

13 

6 

It 

3 

11 

16 

17 

13 

6 

12 

4 

12 

17 

17 

12 

5 

14 

4 

12 

17 

16 

12 

5 

16 

5 

13 

17 

16 

12 

4 

18 

5 

13 

17 

16 

11 

3 

20 

6 

13 

17 

16 

11 

3 

22 

7 

14 

17 

16 

10 

8 

24 

7 

14 

17 

15 

10 

2 

26 

8 

15 

17 

15 

9 

1 

28 

8 

15 

17 

15 

9 

1 

30 

—  9 

—  15 

—  17 

—  15 

—  8 

—  0 

TABLE  XXV. 

Equation  of  the  Sun's  Centre 
Argument.     Sun's  Mean  Anomaly. 


21 


VI* 

VII* 

VIII* 

IX* 

X* 

xu 

11* 

11* 

11* 

11* 

11* 

11* 

o 

O   '    " 

0    '   " 

0   /    " 

O   '    " 

O   '   " 

0    '   ft 

0 

29  59  13.9 

29  235.2 

28  20  23.0 

28  3  53.7 

28  18  17.1 

29  0  29.3 

1 

29  57  15.6 

29  052.4 

28  19  22.2 

28  352.3 

28  19  17.0 

29  2  15.7 

2 

29  55  17.3 

28  59  10.5 

28  18  23.3 

28  3  52.8 

28  20  19.0 

29  4  3.2 

3 

29  53  19.1 

28  57  29.8 

28  1726.1 

28  3  55.6 

28  21  22.7 

29  5  51.8 

4 

29  51  20  9 

28  55  50.0 

28  16  30.7 

28  4  0.5 

•282228.4;  29  741.5 

5 

29  49  23.0 

2854  11.4 

28  1537.1 

28  4  7.4 

28  23  35.8 

29  932.2 

6 

29  47  25.2 

28  52  33.8 

28  14  45.4 

28  4  16.6 

282445.1 

29  11  23.8 

7 

29  45  27.7 

28  50  57.5 

28  13  55.6 

28  427.7 

28  25  56.1 

29  13  16.3 

8 

29  43  30.3 

28  49  22.4 

28  13  7.5 

28  441.0 

28  27  9.0 

29  15  9.7 

9 

29  41  33.2 

28  47  48.5 

28  1221.5 

28  456.4 

28  28  23.6 

29  17  3.9 

10 

29  39  36.4 

28  46  15.7 

28  1  1  37.4 

28  5  13.9 

28  29  39.8 

29  18  59.0 

11 

29  37  39.9 

28  44  44.3 

28  1055.1 

28  533.5 

28  30  57.8 

29  20  54.9 

12 

29  35  43  9 

28  43  14.1 

28  10  14.8 

23  5  55.3 

28  32  17.5 

29  22  51.6 

13 

29  33  48.2 

28  41  45.4 

28  9  36.5 

28  6  19.1 

28  33  38.9 

29  24  48.9 

14 

29  31  53.0 

28  40  17.9 

28  9  0.0 

28  644.9 

2835  1.8 

29  26  46.9 

15 

29  29  58.2 

28  38  51.8 

28  8  25.6 

28  7  12.9 

28  36  26.3 

29  28  45.5 

16 

29  28  4.0 

28  37  27.2 

28  753.2 

28  743.1 

28  37  52.6 

29  30  44.6 

17 

29  26  10.1 

28  36  4.0 

28  722.8 

28  8  15.2 

28  39  20.3 

29  32  44-4 

18 

29  24  17.0 

283442.1 

28  6  54.4 

28  8  49.4 

28  40  49.6 

29  34  44.7 

19 

29  22  24.5 

28  33  21.9 

28  6  28.0 

28  925.6 

28  42  20.3 

29  36  45.4 

20 

29  20  32.5 

28  32  3.0 

28  6  3.6 

28  10  3.9 

28  43  52.5 

29  38  46.6 

21 

29  18  41.3 

28  30  45.8 

28  541.4 

28  10  44.3 

284526.1 

29  40  48.2 

22 

29  16  50.8 

28  29  30.1 

28  521.1 

28  11  26.6 

2847  1.3 

29  42  50.1 

23 

29  15  0.9 

28  28  15.9 

28  5  2.9 

28  12  11.0 

28  48  37.7 

29  44  52.5 

24 

29  13  11.8 

28  27  3.4 

28  446.8 

28  12  57.4 

28  50  15.5 

29  46  55.0 

25 

29  11  23.6 

28  25  52.4 

28  432.6 

28  13  45.7 

28  51  54.8 

29  48  57.8 

26 

29  9  36.2 

28  24  43.2 

28  420.7 

28  14  36.0 

28  53  35.2 

2951  0.8 

27 

29  749.5 

28  23  35.6 

28  4  10.8 

28  15  28.5 

28  55  16.9 

29  53  3.9 

28 

29  6  3.8 

28  22  29.7 

28  4  3.0 

28  16  22.7 

28  56  59.8 

2955  7.2 

29 

29  419.1 

28  21  25.4 

28  3  57.3  '  28  17  18.9 

28  58  43.9 

29  57  10.5 

30 

29  2  35.2 

28  20  23.0  28  3  53.7  28  18  17.1 

29  0  29.3 

29  59  13.9 

TABLE  XXVI. 

Secular   Variation  of  Equation  of  Surfs  Centre. 
Argument.    Sun's  Mean  Anomaly. 


VI* 

VII* 

VIII* 

IX* 

X* 

XI. 

0 

0 

// 

+  0 

+  8 

+  15 

ft 

+  17 

// 

+  15 

+  9 

2 

1 

9 

15 

17 

15 

8 

4 

1 

9 

15 

17 

15 

8 

6 

2 

10 

15 

17 

14 

7 

8 

2 

10 

16 

17 

14 

7 

10 

3 

11 

16 

17 

14 

6 

12 

3 

11 

16 

17 

13 

6 

14 

4 

12 

16 

17 

13 

5 

16 

5 

12 

16 

17 

12 

4 

18 

5 

12 

17 

17 

12 

4 

20 

6 

13 

17 

16 

11 

3 

22 

6 

13 

17 

16 

11 

2 

24 

7 

14 

17 

16 

10 

8 

26 

7 

14 

17 

16 

10 

1 

28 

8 

14* 

17 

15 

9 

1 

30 

+  8 

+  15 

+  17 

+  15 

+  9 

+  0 

30 


22 


TABLE   XXVII. 


Nutations. 
Argument.     Supplement  of  the  Node,  or  N. 


Solar  Nutation. 


N. 

Long. 

R.  Asc. 

Obliq. 

N. 

Long. 

R.  Asc. 

Obliq. 

Long.  Obliq. 

0 

+  0.0 

+0.0 

+  9.2 

500  —  -  0.0 

—  0.0 

—  9.3 

Jan. 

ff 

/> 

10 

1.0 

1.0 

9.1 

510J    1.1 

1.0 

9.3 

1 

+  0.5  —0.5 

20 

2.1 

2.1 

9.1 

520    2.2 

2.0 

9.3 

11 

0.8 

0.4 

30 

3.2 

3.0 

9.0 

530    3.3 

2.9 

9.2 

21 

1.1 

0.2 

40 

4.2 

4.0 

8.9 

540 

4.4 

3.9 

9.0 

31 

1.2 

—  0.1 

50 

+  5.2 

+  4.9 

+  8.7 

550 

—  5.5 

—  4.8 

—  8.9 

Feb. 

60 

6.2 

6.0 

8.5 

560 

6.5 

5.7 

8.7 

10 

1.2 

+  0.1 

70 

7.2 

6.9 

8.3 

570 

7.5 

6.6 

8.4 

20 

1.0 

0.3 

80 

8.2 

7.8 

8.1 

580 

8.5 

7.5 

8.1 

Marrh 

90 

9.1 

8.7 

7.8 

590 

9.5 

8.4 

7.8 

iVLarCIl. 
n 

A  *7 

A  A 

100 

+  10.0 

+  9.4 

+  7.5 

600 

—  10.4 

—  9.1 

—  7.5 

(£ 

12 

U.  . 

+  0.3 

U.^r 

05 

110 

10.8 

10.3 

7.1 

610 

11.2 

9.9 

7.1 

22 

—  0.1 

0.5 

120 

11.6 

11.1 

6.7 

620 

12.0 

10.6 

6.7 

A     '1 

130 
140 
150 

12.4 
13.1 

+  13.8 

11.7 
12.4 
+  13.0 

6.3 

5.9 
+  5.5 

630 
640 
650 

12.8 
13.5 
—  14.2 

11.4 
12.0 
—  12.6 

6.3 
5.9 
—  5.4 

April. 
1 
11 
21 

0.5 
0.8 
1.1 

0.5 
0.2 
0.2 

160 

14.4 

13.6 

5.0 

660 

14.8 

13.2 

4.9 

\r 

170 

15.0 

14.1 

4.5 

670 

15.3 

13.8 

44 

May. 
i 

I  O 

_i_  n  i 

180 
190 

15.5 
15.9 

14.5 
14.8 

4.0 
3.5 

680 
690 

15.8 
16.2 

14.2 
14.7 

3.9 
3.3 

1 

11 

91 

JL  .  A 

1.2 

1  1 

l  U.  1 

—  0.11 

A  q 

200 

+  16.3 

+  15.1 

+  2.9 

700 

—  16.6 

—  15.0 

—  2.8 

/w  1 

31 

1  .  I 

0.8 

v.O 

0.4 

210 

16.6 

15.4 

2.4 

710 

16.9 

15.3 

2.2 

J  unc. 

220 
230 
240 
250 

16.9 
17.1 
17.2 
+  17.3 

15.6 
15.7 
15.9 
+  15.9 

1.8 
1.2 
0.7 

+  0.1 

720 
730 
740 
750 

17.1 
17.2 
17.3 
—  17.3 

15.4 
15.7 
15.9 
—  15.9 

1.6 

1.1 
—  0.5 

+  0.1 

10 
20 
30 

0.4 
—  0.0 

+  0.4 

0.5 
0.5 
0.5 

260 

270 
280 
290 

17.3 
17.2 
17.1 
16.9 

15.9 
15.7 
15.6 
15.4 

—  0.5 

l.l 
1.6 
2.2 

760 
770 
780 
790 

17.2 
17.1 
16.9 
16.6 

15.9 
15.7 
15.4 
15.3 

0.7 
1.2 

1.8 
2.4 

July. 
10 
20 
30 

0.7 
1.0 

1.2 

0.4 
0.3 
—  0.1 

300 

+  16.6 

+  15.1 

_  2.8 

800 

_16.3 

-15.0 

+  2.9 

4ug. 

310 

16.2 

14.8 

3.3 

810 

15.9 

14.7 

3/i 

9 

1  Q 

1.3 

1  2 

+  0.0 

A  A 

320 
330 

158 
15.3 

14.5 
14.1 

3.9 

4.4 

820 
830 

15.5 
15.0 

14.2 
13.8 

4.0 
4.5 

iy 
29 

1  .  £ 

0.9 

U.^r 

0.4 

340 

14.8 

13.6 

4.9 

840 

14.4 

13.2 

5.0 

Sept. 

350 

+  14.2 

4.  13.0 

_  5.4 

850 

__13.8 

—  12.6 

+  5.5 

8 

0.6 

0.5 

360 
370 

13.5 

12.8 

12.4 
11.7 

5.9 
6.3 

860 
870 

13.1 
12.4 

12.0 
11.4 

5.9 
6.3 

18 
28 

+  0.2 
—  0.2 

0.5 

0.5 

380 

12.0 

11.1 

6.7 

880 

11.6 

10.6 

6.7 

Oct. 

390 

11.2 

10.3 

7.1 

890 

10.8 

9.9 

7.1 

8 

0.6 

0.5 

400 

+  10-4 

+  9.4 

—  7.5 

900 

—  10.0 

—  9.1 

+  7.5 

18 

1.0 

0.3 

OO 

1  O 

0  2 

410 

9.5 

8.7 

7.8 

910 

9.1 

8.4 

7.8 

ivO 

I  .& 

420 

85 

7.8 

8.1 

920 

8.2 

7.5 

8.1 

Nov. 

430 

7.5 

6.9 

8.4 

930 

7.2 

6.6 

8.3 

7 

1.2 

+  0.0 

440 

6.5 

6.0 

8.7 

940 

6.2 

5.7 

8.5 

17 

1.2 

0.2 

450 

+  5.5 

+  4.9 

_  8.9 

950 

—  5.2 

—  4.8 

+  8.7 

27 

1.0 

0.4 

460 

4.4 

4.0 

9.0 

960 

4.2 

3.9 

8.9 

Dec. 

470 

3.3 

3.0 

9.2 

970 

3.2 

2.9 

9.0 

7 

0.6 

0.5 

480 

2.2 

2.1 

9.3 

980 

2.1 

2.0 

9.1 

17 

—  0.2 

0.5 

490 

1.1 

1.0 

9.3 

990 

1.0 

1.0 

9.1 

27 

+  0.3 

0.5 

500 

+  0.0 

+  0-0  —  9.3 

1000 

—  0.0 

—  0.0 

+  J>.2 

37 

+  0.6 

—  0.5 

TABLE   XXVIII. 


TABLE  XXIX 


Lunar  Equation, 
Argument  I. 


Lunar  Equation,  2d  part. 
Arguments  I.  and  VI. 
I. 


I  |Equ 

I   Equ 

i 

j 

0 

7.5 

500  7.5 

10 

8.0 

510  7.0 

20 

8.4 

520  6.6 

30 

8.9 

530,6.1 

40 

9.4 

540  5.6 

50 

9.8 

550  5.2 

60 

10.3 

560  !  4.7 

70 

10.7 

570  4.3 

80 

11.1 

580  3.9 

90 

11.5 

590  3.5 

100 

11.9 

600  3.1 

110 

12.3 

610 

2.7 

120 

12.6 

620 

2.4 

130 

13.0 

630 

2.0 

140 

13.3 

640 

1.7 

150 

13.6 

650 

1.4 

160 

13.8 

660 

1.2 

170 

14.1 

670 

0.9 

180 

14.3 

680 

0.7 

190 

14.5 

690 

0.5 

200 

14.6 

700 

0.4 

210 

14.8 

710 

0.2 

220 

14.9 

720 

0.1 

230 

14.9 

730 

0.1 

240 

15.0 

740 

0.0 

250 

J5.0 

750 

0.0 

260 

15.0 

760 

0.0 

270 

14.9 

770 

0.1 

280 

14.9 

780 

0.1 

290 

14.8 

790 

0.2 

300 

14.6 

800 

0.4 

310 

14.5 

810 

0.5 

320 

14.2 

820 

0.7 

330 

14.1 

830 

0.9 

340 

13.8 

840  i  1.2 

350 

13.6 

850 

1.4 

360 

13.3 

860 

1.7 

370 

13.0 

870 

2.0 

380 

12.6 

880 

2.4 

390 

12.3 

890 

2.7 

400 

11.9 

900 

3.1 

410 

11.5 

910 

3.5 

420 

11.1 

920 

3.9 

430 

10.7 

930 

4.3 

440 

10.3 

940  4.7 

450 

9.8 

950  i  5.2 

460 

9.4 

960 

5.6 

470 

8.9 

970  6.1 

480 

84 

980  6.6 

490 

80 

990  !  7.0 

500   75 

000  7.5 

VI 

0 

50 

100 

150  200 

250 

300 

'350 

•400 

450 

GOO 

0 

1.3 

1.2 

1.2 

1.1 

1.0 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

50  1.5 

1.5 

1.5 

1.3 

1.1 

1.0 

0.9 

1.0 

1.1 

1.1 

1.1 

100  1.7 

1.8 

1.7 

1.4 

1.2  1.1 

1.0 

0.9 

0.9 

0.9 

0.9 

150  i  1.9 

1.9 

1.8 

1.6 

1.4 

1.3 

1.0 

0.8 

0.8 

0.8  0.7 

200 

1.9 

2.0 

2.0  1.7 

1.5 

1.4 

1.0 

0.8 

0.8 

0.8  0.7 

250 

2.0 

2.0  2.0 

1.8 

1.6 

1.5 

1.1 

0.9 

0.7 

0.7  0.6 

300 

1.9 

1.9  1.9 

1.9 

1.7 

1.6 

1.2 

1.0 

0.8 

0.7 

0.7 

350 

1.8 

1.9 

1.9 

1.9 

1.7 

1.6 

1.4 

1.0 

1.0 

0.9 

0.8 

400 

1.6 

1.7J1.8 

1.9 

..7 

1.6 

1.4 

1.2 

l.ljl.O 

1.0 

450 

1.5 

1.5 

1.6 

1.7 

1.7 

1.7 

1.6 

1.4 

1.2 

1.2 

1.1 

500 

1.3 

1.4 

1.4 

1.5 

1.7 

1.7 

1.7 

1.5 

1.4 

1.4 

1.3 

550 

1.1 

1.2 

1.2 

1.4 

1.6 

1.7 

l.V 

i  7 

1.6 

1.5 

1.5 

600 

1.0 

1.0 

1  1  1.2 

1.4 

1.6 

1.8 

1.8  1.8 

1.7 

1.6 

650 

0.8 

0.9 

1.0 

1.1 

1.3 

1.5 

1.7 

1.8  1.9 

1.9 

1.8 

700 

0.7 

0.7 

0.8 

1.1 

1  1  2 

1.4 

1.7 

1.9  1  9 

1.9 

1.9 

75U 

0.6 

0.6 

0.7 

1.0 

LI 

1.3 

1.6 

1.9  1.9 

2.0 

2.0 

800 

0.7 

0.7 

0.7 

0.9 

1.1 

1.2 

1.5 

1.8  2.0 

1.9 

1.9 

850 

0.7 

0.8 

0.8 

0.9 

0.9 

1.1 

1.4 

1.7  1.8 

1.8 

1.9 

900 

0.9 

0.9 

0.9 

0.9 

1.0 

1.1 

1.2 

1.5  1.7 

1.7 

1.7 

950 

1.1 

1.0 

1.1 

1.0 

1.0 

1.0 

1.1 

1.3  1.4 

1.6 

1.5 

0 

1  3 

1.2  1.2 

1.1 

1.0 

1.0 

1.0 

1.1.1.2 

1.2 

1.3 

I. 

VI 

500 

550 

600 

650 

700 

750 

800 

850 

900 

950  1000 

0 

1.3 

1.4 

1.4 

1.5 

1.6 

1.6 

.6 

1.5 

.4 

1.4  .3 

50 

1.1 

1.1 

1.2  1.3 

1.5 

1.5 

.7 

1.6 

.5 

1.5   .5 

100 

0.9 

0.9 

0.9  1.1 

1.3 

1.5 

.6 

1.7 

.7 

1.7   .7 

150 

0.7 

0.8 

0.8  0.9 

1.2 

1.4 

.6 

1.9 

.8 

1.8   .9 

200 

0.7 

0.7 

0.6  0.8 

1.1 

1.2 

.6 

1.8 

.8 

1.8 

.9 

250 

06 

06 

0.7  0.7 

1  0 

1  1 

1  5 

1  7 

9 

;  9 

20 

300 

0.7 

0.7 

0.7 

0.7 

0.9 

1.0 

1.4 

1.6 

.8 

1.9 

.9 

350 

0.8 

0.7 

0.7 

0.8 

0.9 

1.0 

1.4 

1.6 

.6 

1.7 

.8 

400 

1.0 

0.9 

0.8 

0.8 

0.9 

1.0 

1.2 

1.4 

.5 

1.6 

.6 

450 

1.1 

1.1 

1.0 

0.9 

0.9 

0.9 

1.0 

1.2 

.4 

1.4 

1.5 

500 

1.3 

1.2 

1.2 

1.110.9 

0.9 

0.9 

1.1 

.2 

1.2 

1.3 

550 

1.5 

.4 

1.4 

1.2 

1.0 

0.9 

0.9 

0.9 

1.0 

1.1 

1.1 

600 

1.6 

.6 

1.5 

1.4 

1.2 

1.0 

0.8 

0.8 

0.8 

0.9 

1.0 

65C  1.8  1  .7 

1.6 

1.6 

1.3 

1.1 

0.9 

0.8 

0.7)0.7 

0.8 

700  1.9 

.8 

1.8 

1.6 

1.4 

1.2 

0.9 

0.7 

0.7  0.7 

0.7 

750  2.0 

.9 

1.9 

1.7 

1.5 

1.3 

1.0 

0.7 

0.7  0.6 

0.6 

800  1.9)1.8 

1.8 

1.8 

1.6 

1.4 

1.1 

0.8 

0.6  0.7 

0.7 

850  1.9  1.8 

1.8 

1.8 

1.6 

1.5 

1.2 

0.9 

0.8  0.8 

0.7 

90011.7  1.7 

1.7 

1.7 

1.6 

1.5 

1.3 

1.1 

0.9  0.9 

0.9 

950:  1.5  1.5 

1.5 

1.6 

1.7 

1.6  1.5 

1.3 

1.2  1.1 

1.1 

Oil.3 

1.4  1.4 

1.5 

1.6 

1.6  1.6 

1.5 

1.4  1  1.4 

1.3 

Constant  1".3. 

24: 


TABLE  XXX. 


Perturbations  produced  by   Venus. 

Arguments  II  and  III. 

HI. 


II. 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

0 

" 
21.6 

20.8 

19.8 

19.0 

17.9 

16.8 

15.9 

14.7 

14.0 

13:2 

128 

125 

12.3 

20 

23.1 

22.7 

21.6 

21.0 

20.1 

19.3 

18.4 

17.4 

16.4 

15.5 

14.5 

13.8 

13.4 

40 

23.5 

23.2 

22.9 

22.7 

22.0 

21.1 

20.4 

19.6 

18.7 

17.9 

16.9 

16.1 

15.3 

60 

22.2 

22.5 

23.1 

22.7 

22.8 

22.5 

21.9 

21.3 

20.5 

19.9 

19.1 

18.2 

17.4 

80 

20.0 

20.7 

21.4 

21.7 

22.1 

22.3 

22.2 

22.2 

21.7 

21.3 

20.7 

19.9 

19.3 

100 

17.6 

18.6 

19.2 

19.9 

20.5 

21.0 

21.6 

21.7 

21.6 

21.6 

21.5 

21.1 

20.5 

120 

15.3 

16.0 

16.9 

17.7 

18.4 

19.2 

19.8 

20.2 

20.7 

20.8 

21.1 

21.1 

20.8 

J40 

13.6 

14.2 

14.8 

15.5 

16.2 

17.0 

17.6 

18.3 

19.0 

19.4 

20.0 

20.0 

20.4 

160 

12.7 

13.2 

13.6 

14.1 

14.6 

15.0 

15.7 

16.4 

17.0 

17.3 

18.1 

18.7 

19.2 

180 

12.7 

12.9 

13.1 

13.5 

13.9 

14.0 

14.5 

14.8 

15.0 

15.8 

16.4 

16.8 

17.2 

200 

13.2 

13.2 

13.2 

13.4 

13.7 

13.8 

14.1 

14.2 

14.5 

14.5 

14.8 

15.2 

16.0 

- 

220 

13.5 

13.6 

13.9 

14.1 

14.1 

14.1 

14.2 

14.3 

14.5 

14.6 

14.6 

14.7 

14.8 

240 

13.6 

13.8 

14.1 

14.4 

14.6 

14.8 

14.8 

14.9 

15.1 

15.1 

15.1 

14.9 

14.8 

260 

12.8 

13.3 

13.8 

14.2 

14.6 

15.0 

15.3 

15.6 

15.5 

15.5 

15.6 

15.6 

15.6 

280 

11.5 

12.3 

13.0 

13.4 

14.0 

14.6 

15.1 

15.4 

16.0 

16.2 

16.2 

16.3 

16.2 

300 

10.1 

10.9 

11.3 

12.1 

12.9 

13.7 

14.2 

14.9 

15.4 

16.0 

16.4 

16.5 

16.7 

320 

8.2 

8.8 

9.6 

10.6 

11.3 

12.0 

12.9 

13.7 

14.3 

15.0 

15.8 

16.3 

16.8 

340 

6.9 

7.5 

8.1 

8.4 

9.4 

10.1 

11.1 

11.9 

12.7 

13.6 

14.4 

15.2 

16.0 

360 

6.5 

6.5 

6.8 

7.4 

.8.0 

8.4 

9.1 

9.9 

10.8 

11.5 

12.6 

13.4 

14.4 

380 

6.8 

6.5 

6.3 

6.4 

6.7 

7.0 

7.6 

8.2 

8.9 

9.6 

10.6 

11.4 

12.4 

400 

7.5 

7.1 

6.7 

6.4 

6.2 

6.4 

6.5 

6.9 

7.5 

7.9 

8.7 

9.4 

10.3 

420 

9.1 

8.4 

7.6 

7.1 

6.7 

6.5 

6.3 

6.2 

6.7 

6.8 

7.2 

7.8 

8.4 

440 

10.6 

9.8 

9.0 

8.6 

7.9 

7.2 

6.7 

6.4 

6.4 

6.4 

6.6 

6.8 

7.1 

460 

12.1 

11.5 

10.5 

9.6 

9.0 

8.5 

8.0 

7.3 

6.8 

6.6 

6.5 

6.4 

6.5 

480 

13.6 

12.8 

11.9 

11.0 

10.4 

9.6 

8.8 

8.2 

7.7 

7.2 

6.8 

6.4 

6.5 

500 

15.1 

14.4 

13.4 

12.4 

11.6 

10.8 

10.1 

9.3 

8.6 

8.1 

7.5 

7.1 

6.8 

520 

16.5 

15.6 

14.8 

13.9 

13.1 

12.3 

11.3 

10.5 

9.7 

9.1 

8.6 

7.9 

7.4 

540 

18.1 

17.5 

16.4 

15.5 

14.5 

13.7 

12.8 

11.8 

11.1 

10.4 

9.7 

8.9 

8.2 

560 

20.4 

19.3 

18.2 

17.6 

16.5 

15.4 

14.4 

13.4 

12.7 

11.6 

10.8 

10.2 

9.2 

580 

22.8 

21.7 

20.7 

19.7 

18.4 

17.6 

16.6 

15.5 

14.3 

13.4 

12.5 

11.6 

10.6 

600 

25.2 

24.1 

23.1 

22.2 

21.2 

19.9 

18.6 

17.8 

16?6 

15.6 

14.5 

13.4 

12.6 

620 

27.3 

26.5 

25.6 

24.7 

23.5 

22.5 

21.6 

20.4 

19.0 

18.1 

16.8 

15.7 

14.7 

640 

29.0 

28.5 

27.7 

26.9 

26.2 

25.1 

24.1 

22.9 

21.8 

20.8 

19.6 

18.4 

17.2 

660 

29.8 

29.6 

29.2 

28.5 

28.1 

27.4 

26.5 

25.6 

24.5 

23.4 

22.5 

21.2 

19.8 

680 

29.7 

29.6 

29.5 

29.5 

29.1 

28.8 

28.2 

27.6 

27.0 

26.0 

25.0 

23.8 

22.8 

700 

28.8 

29.2 

29.3 

29.5 

29.5 

29.5 

29.2 

28.8 

28.4 

27.8 

27.2 

26.4 

25.2 

720 

26.9 

27.6 

28.3 

29.0 

29.2 

29.4 

29.4 

29.3 

29.1 

28.9 

28.4 

27.9 

27.3 

740 

24.7 

25.7 

26.6 

27.3 

27.9 

28.5 

29.1 

29.0 

29.2 

29.3 

29.1 

28.8 

28.4 

760 

22.2 

23.5 

24.3 

25.3 

26.2 

27.0 

27.6 

28.3 

28.6 

28.7 

28.9 

29.1 

29.0 

780 

19.6 

21.0 

22.0 

23.2 

24.2 

25.1 

25.9 

26.7 

27.3 

27.8 

28.4 

28.5 

28.7 

800 

17.2 

18.5 

19.3 

20.9 

21.8 

22.9 

23.9 

25.0 

25.8 

264 

26.9 

27.6 

28.1 

820 

15.2 

15.9 

17.0 

18.4 

18.9 

20.7 

21.7 

22.8 

23.8 

24.8 

25.6 

26.2 

26.6 

840 

13.2 

14.0 

150 

16.0 

17.0 

18.2 

18.8 

20.3 

21.7 

22.7 

23.6 

24.5 

25.3 

860 

11.5 

12.2 

13.0 

139 

14.9 

15.9 

17.1 

18.0 

18.9 

20.3 

21.4 

22.6 

23.5 

880 

11.0 

11.2 

11.5 

12.2 

13.0 

13.7 

14.8 

15.7 

16.8 

18.1 

19.1 

20.2 

21.1 

900 

11.2 

10.2 

10.9 

11.5 

12.5 

12.1 

12.8 

13.7 

14.5 

15.5 

16.6 

17.9 

18.5 

920 

12.1 

11.6 

11.5 

11.1 

11.2 

11.3 

11.7 

12.1 

12.7 

13.4 

144 

15.2 

16.4 

940 

14.0 

13.3 

12.6 

12.3 

11.6 

11.5 

11.3 

11.4 

11.6 

12.0 

12.8 

13.3 

14.2 

960 

16.7 

15.6 

14.6 

13.7 

13.1 

12.5 

11.9 

11.7 

11.6 

11.4 

11.7 

12.1 

12.6 

980 

19.5 

18.3 

17.3 

16.4 

15.2 

14.2 

13.4 

12.7 

lfc.2 

12.0 

11.9 

11.8 

11.8 

1000 

21,6 

20.8 

19.8 

19.0 

17.9 

16.8 

15.9 

14.7 

14.0 

13.2 

12.8 

12.5 

12.2 

0 

10 

20 

30 

40 

59 

60 

70 

80 

90 

00 

110 

120 

TABLE  XXX. 


25 


Perturbations  produced  by   Venus. 

Arguments  II  and  III. 

III. 


II.   120  130  140  150  160  170 

180  190 

200 

210 

220 

230 

240 

I 

0  I  12.2  12.2  12.3  12.4  12.8  •  13.3 

13.9  14.7  15.6 

16.5 

17.7 

18.8 

20.1 

20  ;  13.4  12.9  ;  12.6 

12.3  12.2  12.4 

12.9  13.3  14.0 

14.6 

15.5 

16.4 

17.3 

40  15.3  14.4  14.0 

13.5  13.0  12.9 

12.6  j  12.6  13.1 

13.5 

14.0 

14.4 

15.4 

60  17.4  16.7 

16.0 

15.2  14.5  14.0 

13.6 

13.3  13.2  i  13.2 

13.4 

13.5 

14.1 

80  19.3  18.7 

17.7 

17.1  16.4(  15.9 

15.4 

14.6  14.3 

13.9 

13.8 

13.7 

13.6 

100  20.5  20.2 

19.5 

18.9  18.2 

17.5 

17.1 

16.3  .  15.9 

15.4 

14.8 

14.6 

14.3 

;         I 

120  20.8  20.7 

20.4 

20.0  ;  19.7 

192 

18.5 

18.0'  17.3 

16.9 

16.5 

16.2 

15.6 

140  20.4  20.4 

20.2 

20.0  20.1 

19.7 

19.5 

19.3 

18.8 

18.2 

17.7 

17.4 

17.0 

160 

19.2  19.1 

19.4 

19.7  19.5 

19.6 

19.3 

19.6 

19.2 

19.0 

18.7 

18.4 

18.1 

180 

17.2 

17.7 

18.5 

18.5  !  18.5 

18.8 

18.4  18.8 

19.0 

19.0 

18.9 

18.6 

18.5 

200 

16.0 

16.2 

16.6 

16.8 

17.5 

17.6 

17.7 

17.9 

18.1 

18.2 

18.3 

18.3 

18.3 

220 

14.8 

15.0 

15.3 

15.7 

16.1 

16.2 

16.6 

16.8 

17.1 

17.5 

17.1 

17.4 

17.5 

240 

14.8 

14.7 

14.8 

15.0 

15.1 

15.4 

15.7 

15.8 

16.0 

16.1 

16.1 

16  ° 

16.1 

260 

15.6 

15.7 

15.3 

14.8 

15.0 

15.0 

15.1 

15.0 

15.1 

15.2 

15.2 

15.1 

15.3 

280 

L«.2 

16.2 

16.2  15.9 

15.8 

15.8 

15.5 

15.4 

15.1 

14.9 

14.8 

14.7 

15.0 

300 

16.7 

17.0 

17.1 

16.9 

16.9 

16.6 

16.5 

16.3 

15.9 

15.7 

15.2 

14.9 

14.8 

320 

16.8 

17.3 

17.5 

17.6 

17.7 

17.6 

17.5 

17.2 

17.0 

16.8 

16.5 

16.1 

15.6 

340 

16.0 

16.4 

17.2 

IV.  8 

17.9 

18.1 

18.3 

18.2 

18.2 

17.9 

17.5 

17.3 

16.8 

360 

14.4 

15.2 

16.0 

16.7 

17.4 

18.1 

18.4 

18.6 

18.8 

18.8 

18.8 

18.7 

18.4 

380 

12.4 

13.4 

14.3 

15.3 

16.1 

16.9 

17.5 

18.1 

18.6 

19.1 

19.3 

19.5 

19.5 

400 

10.3 

11.2 

12.3 

13.2 

14.2 

15.1 

16.0 

16.8 

17.8 

18.4 

18.8 

19.3 

19.8 

420 

8.4 

9.2 

10.0 

11.0 

12.2 

13.0 

14.1 

15.0 

15.9 

16.9 

17.7 

18.5 

19.0 

440 

7.1 

7.6 

8.4 

9.0 

9.9 

10.9 

11.8 

12.9 

13.8 

14.9 

16.0 

16.7 

17.8 

460 

6.5 

6.8 

7.2 

7.4 

8.1 

9.0 

9.7 

10.6 

11.7 

12.6 

13.8 

14.6 

15.9 

480 

6.5 

6.5 

6.4 

6.6 

7.0 

7.5 

8.2 

8.8 

9.6 

10.4 

11.5 

12.5 

13.5 

500 

6.8 

6.7 

6.5 

6.3 

6.5 

6.6 

7.0 

7.4 

8.2 

8.6 

9.4 

10.4 

11.3 

520 

7.4 

7.0 

6.8 

6.5 

6.3 

6.1 

6.3 

6.6 

7.0 

7.5 

8.0 

8.8 

9.3 

540 

8.2 

7.6 

7.2 

6.8 

6.5 

6.3 

6.2  i  6.0 

6.2 

6.5 

6.9 

7.4 

7.9 

560 

9.2!  8.6 

7.9 

7.5 

6.8 

6.6 

6.3  1  6.1 

6.0 

6.1 

6.2 

6.5 

6.9 

580 

10.6 

9.8 

9.1 

8.4 

7.7 

7.3 

6.6  !  6.3 

6.1 

5.9 

5.7 

5.9 

6.0 

600 

12.6 

11.4 

10.5 

9.5 

8.7 

8.1 

7.4 

7.0 

6.4 

6.1 

5.8 

5.5 

5.6 

G20 

14.7 

13.5 

12.4 

11.4 

10.4 

9.5 

8.7 

7.9 

7.3 

6.7 

6.2 

5.6 

5.2 

040 

17.2 

16.2 

14.9|  13.7 

12.5 

11.4 

10.4 

9.5 

8.7 

7.8 

7.0 

6.5 

5.9 

600  198 

19.0  1  17.6  |  16.5 

15.1 

13.9 

12.8 

11.5 

10.5 

9.6 

8.6 

7.7 

6.9 

680  22.8 

21.7 

20.4 

19.3 

18.1 

16.8 

15.7 

14.2 

13.0 

11.9 

10.7 

9.6 

8.6 

700  25.2  24.3 

23.3 

22.1 

20.7 

19.7 

18.5 

17.3 

160 

14.3 

13.4 

12.1 

11.0 

720  ''27.3  26.4i25.7  24.5 

740  I  28  4  1  27.  7  27.4)26.6 

23.7 
25.9 

22.5 
24.9 

21.1 
24.0 

20.2 

22.8 

18.8 
21.5 

17.7 
20.6 

16.4 
19.2 

15.3 

18.1 

13.9 
16.8 

760  J29.0  28.7 

28.3 

27.8 

27.3 

26.8 

25.9 

25.2 

24.3 

23.0 

21.7 

20.7 

19.7 

780  !  28.7  28.7 

28.8 

28.7 

28.3 

28.0 

27.2 

26.1 

26.1 

25.2 

24.3 

23.3 

22.2 

800 

28.  l!  28.3 

28.4 

28.5 

28.5 

28.4 

28.2 

27.3 

27.3 

26.7 

25.9 

25.1 

24.4 

820 

26.6  27.3 

27.8 

28.1 

28.3 

28.1 

28.1 

28.0 

27.9 

27.7 

27.2 

26.5 

25.9 

840 

25.3  26.2  26.7 

27.2 

27.5 

27.9 

28.1 

28.1 

27.9 

27.9 

27.6 

27.3 

27.2 

860 

23.5 

24.5 

25.1 

25.9 

26.6 

27.1 

27.4 

27.7 

27.9 

28.0 

27.9 

27  7 

27.5 

880 

21.1 

22.4 

23.3 

24.2 

25.1 

25.8 

26.5 

27.0 

27.3 

27.5 

27.8 

280 

27.7 

900 

18.5 

20.1 

21.3 

22.1 

23.1 

24.7 

25.0 

25.7 

26.3 

26.9 

27.3 

27.5 

27.6 

920 

16.4 

17.7 

18.4 

20.0 

21.0 

22.2 

23.0 

23.9 

24.9 

25.7 

26.2 

26.9 

27.3 

940 

14.2 

14.9 

16.1 

17.5  18.2 

19.6 

20.8 

21.9 

23.0 

23.9 

24.7 

25.7 

26.1 

960 

12.6 

13.3 

14.1 

14.4 

15.9 

17.2 

17.9 

19.5 

20.5 

21.7 

22.7 

23.9 

24.7 

980 

11.8 

12.1 

12.7 

13.3  14.1 

14.8 

15.6 

16.8 

17.6 

19.3 

20.2 

21.4 

22.6 

1000 

12.2 

12.2 

12.3 

12.4  12.8 

13.3 

13.9 

14.7 

15.6 

16.5 

17.6 

18.8 

20.1 

20  130  140 

150  160 

170 

ISO  190 

200 

210 

220 

230 

24> 

26 


TABLE  XXX. 


Perturbations  produced  by   Venus. 

Arguments  II.  arid  III. 

HI. 


II. 

240 

250 

260 

270  280 

290 

300 

310 

320 

330 

340 

350  i  360 

0 

20.1 

21.1 

22  2 

23.4  24.3 

25.2 

25.8 

26.6 

27.2 

27.6 

27.7 

27.6 

27.6 

20  17.3 

18.6 

19.7 

20.9  21.9 

23.0 

24.2 

24.9 

25.8 

26.6 

27.0 

27.4  27.7 

40  15.4 

16.5 

17.3 

18.3  19.4 

20.5 

21.6 

22.7 

23.7 

24.9 

25.5 

26.3  26.9 

60'  14.1 

14.6 

15.2 

16.3  17.2 

18.1 

1S.9 

20.3 

21.2 

22.3 

23.4 

24.5  25.3 

ao  1  13.6 

14.0 

14.5 

14.9  15.5 

16.3 

17.3 

18.2 

19.0 

20.0 

21.1 

22.0;  23.1 

ii>9  14.3 

14.3 

14.3 

14.4  14.6 

15.0 

15.5 

16.2 

16.9 

17.7 

18.9 

19.8  20.8 

120  15.6 

15.2 

14.8 

14.8  15.0 

14.9 

15.0 

15.2 

15.9 

16.3 

17.0 

17.7  18.5 

i40  17.0 

16.6 

16.4 

15.8  15.5 

15.4 

15.6 

15.6 

15.5 

15.6 

16.1 

16.7  j  17.1 

•r-V  18.1 

17.7 

17.5 

17.3  16.9 

16.6 

16.3 

15.9 

16.1 

16.3 

16.3 

16.2  |  16.5 

lft'0  ;  18.5 

18.5 

18.3 

18.1  i  17.9 

17.6 

17.5 

17.3 

17.0 

16.9 

16.7 

16.8  16.9 

200  18.3 

18.4 

18.2 

18.2  ,  18.2 

18.2 

18.1 

18.1 

17.8 

17.7 

17.6 

17.5 

17.7 

220 

17.5 

17.6 

17.8 

17.8 

18.0 

18.0 

18.2 

18.1 

18.1 

18.3 

18.4 

18.3 

18.3 

240 

164 

16.5 

16.7 

16.9 

17.1 

17.3 

17.3 

17.7 

17.5 

18.0 

18.3 

18.4 

18.6 

2GO  15.3 

15.5 

15.5 

15.6  15.8 

16.1 

16.4 

16.6 

16.8 

16.9 

17.4 

17.7 

18.2 

280  15.0 

14.9 

14.9 

14.9  i  14.9 

14.7 

15.0 

15.3 

15.5 

15.9 

16.1 

16.4 

16.8 

300  14.8 

14.6 

14.6 

14.2 

14.0 

14.0 

13.9 

13.9 

14.2 

14.5 

14.8 

15.0 

15.5 

320  15.6 

15.3 

14.7 

14.5 

14.4 

13.1 

13.6 

13.4 

13.3 

13.1 

13.4 

13.6 

13.8 

340 

16.8 

16.6 

16.0 

15.5 

15.2  i  14.5 

14.3 

13.7 

13.1 

13.0 

12.7 

12.6 

12.6 

360 

18.4 

17.9 

17.5 

17.0 

16.5 

15.9 

15.4 

14.9 

14.3 

13.7 

13.0 

12.6 

12.3 

380 

19.5 

19.2  18.9 

18.5 

17.9 

17.7 

16.9 

16.4 

15.8 

15.0 

14.5 

13.6 

13.1 

400 

19.8 

19.8 

20.1 

19.7 

19.4 

19.1 

18.6 

18.1 

17.5 

17.0 

16.1 

15.2 

14.8 

420 

19.0 

19.5 

20.0 

20.3 

20.3 

20.3 

20.1 

19.4 

19.0 

18.9 

18.1 

17.3 

16.5 

440 

17.8 

18.7 

19.2 

19.7 

20.1 

20.4 

20.7 

20.7 

20.5 

20.2 

19.8 

19.5  18.6 

460 

15.9 

16.8 

17.6 

18.6 

19.2 

19.9 

20.3 

20.6 

21.0 

20.9 

20.9 

20.8  20.3 

480  I  13.5 

14.6 

15.5 

16.6 

17.7 

18.5 

19.3 

19.9 

20.5 

20.8 

21.1 

21.2 

21.2 

500 

11.3 

12.4 

13.4 

14.4 

15.5 

15.5 

17.7 

18.6 

19.1 

19.9 

20.7 

21.0 

21.4 

520 

9.3 

10.2 

11.2 

12.2 

13.3  14.2 

15.4 

16.4 

17.6 

18.4 

19.2 

19.8 

20.6 

540 

7.9 

8.6 

9.4 

10.1 

11.  li  12.1 

13.1 

142 

15.3 

16.3 

17.4 

18.3 

19.2 

560 

6.9 

7.2 

7.8 

8.4 

9.2110.1 

11.0' 

11.9 

13.1 

14.1 

15.2 

16.2 

17.2 

580 

6.0 

6.3 

6.6 

7.0 

7.6!  8.4 

9.1 

99 

10.9 

11.9 

12.9 

14.1 

15.0 

600 

5.6 

5.6 

5.8 

6.1 

65  6.8 

7.4 

8.1 

8.8 

9.9 

10.7 

11.8 

12.8 

620 

5.2 

5.4 

5.3 

5.3 

5.5  5.9 

6.3 

6.6 

7.2 

8.0 

8.7 

9.5 

10.6 

640 

5.9) 

5.6 

5.2 

4.9 

5.0  5.0 

5.2 

5.5 

5.8 

6.4 

7.0 

7.6 

8.5 

660 

6.9 

6.3 

5.7 

5.4 

5.0  4.8 

4.5 

4.7 

4.9 

5.1 

5.5 

6.0 

6.8 

680 

8.6 

7.6 

6.Q 

6.2 

5.6  5.1 

4.8 

4.6 

4.2 

4.2 

4.5 

4.6 

5.1 

700 

11.0 

10.0 

s.r 

7.8 

6.8   6.3 

5.6 

5.0 

4.6 

4.2 

4.2 

4.0 

4.2! 

720 

13.9 

12.5 

11.2 

10.3 

9.1   7.9 

7.1 

6.2 

5.6 

4.8 

4.5 

4.2 

3.8  i 

740 

16.8 

15.5 

14.4 

130 

11.7  10.5 

9.4 

8.4 

7.2 

6.5 

5.6 

5.0 

4  3! 

760 

19.7 

185 

17.2 

15.9 

14.7  13.5 

12.2 

10.8 

9.8 

8.9 

7.6 

6.7 

b.9 

780 

22.2 

21.2 

20.1 

19.0 

17.6 

16.3 

15.1 

14.0 

12.6 

11.6 

10.2 

9.2  1  8.1 

800 

24.4 

23.4 

22.2 

21.3 

20.3 

19.2 

18.0 

16.7 

15.4 

14.3 

13.2 

11.9 

10.8 

820 

25.9 

25.1 

24.4 

23.3 

22.3 

21.6 

20.4 

19.4 

18.2 

17.2 

15.9 

14.6 

13.6 

840 

27.2 

26.6 

25.8 

25.0 

24.3 

23.5 

22.4 

21.  6!  20.5 

19.4  18.4 

17.3 

16.4 

860 

27.5 

27.1 

26.8 

26.4 

25.5 

24.8 

24.3 

23.3  22.2 

21.5  20.5 

19.6 

18.4 

880  I  27.7 

27.5 

27.2 

27.0 

2G.5 

26.0 

25.5 

24.7 

24.1 

23.2  22.0 

21.4 

20.4 

900 

27.6 

27.8 

27.9 

27.6 

27.1 

26.7 

26.5125.7 

25.3 

24.6  j  23.9  23.0 

22.0 

920 

27.3 

27.5 

27.5 

27.6 

27.7 

27.5 

27.2  26.7 

26.3 

25.7  '  25.1 

24.3 

23.6 

940 

26.1 

26.7 

27.2 

27.4 

27.7 

27.7 

27.6  27.5127.1 

26.6  26.2 

25.6 

25.5 

960 

24.7 

25.4 

26.2 

26.6 

27.2 

27.5 

27.7  27.7  27.6 

27.4  27.1 

27.0 

26.2 

98Q 

22.6 

23.7 

24.6 

25.3 

25.9 

26.8 

27.2  i  27.5  i  27.7 

27.8  27.6 

27.5 

27.1 

1000 

SO.l 

21.1 

22.2 

23.4 

24.3 

25.2 

25.8  26.6  27.2 

27.6  27.7  27.6 

27.6 

240 

250 

260 

270 

280 

290 

300 

310  320 

330  340  350 

300 

TABLE  XXX. 


27 


Perturbations  produced  by   Venus. 

Arguments  II.  and  III. 

III. 


II. 

360  370 

380 

390 

400 

410  420 

430 

440  ;  450 

460 

470  480 

i 

[ 

i 

0 

27.6 

27.7 

27.3 

26.7 

26.2 

25.5  24.7 

23.8 

23.1 

22.3 

21.3 

20.2 

19.3 

20 

27.7  27.8 

27.8  1  27.6 

27.4  j  26.8  26.2 

25.6 

24.8  24.0 

23.1 

22.0 

20.9 

40 

26.9  273 

27.6  '  27.9 

27.9 

27.7  27.5 

27.1 

26.3  25.6 

24.9 

24.0 

23.2 

60 

25.3 

26.0 

26.8127.1 

27.5 

27.9  27.8 

27.7 

27.3,27.1 

26.7 

25.9  250 

80 

23.1 

24.0 

25.1 

25.9 

26.5 

27.3  27.5 

27.9 

28.2  28.0 

27.6 

27.5  27.2 

100 

20.8  I  21.8 

22:6 

23.6 

24.6 

25.5 

26.2 

26.7 

27.2 

27.5 

27.6 

27.8  j  27.4 

120 

18.5  19.6 

20.6 

21.5 

22.4 

23.2  24.1 

25.1 

25.8 

26.4 

26.9 

27.3  27.5 

140 

17.1  17.9 

18.6 

19.3 

20.3 

21.3  |23.0 

22.9 

23.7 

24.7 

25.5 

26.0  |  26.7 

160 

16.5  17.1 

17.4 

18.1 

18.8 

19.3  20.1 

21.0 

21.9 

22.6 

23.5 

24.2  25.1 

180 

16.9 

17.0 

17.1 

17.4 

18.0 

18.4 

18.9 

19.4 

20.1 

20.7 

21.2 

22.2 

23.0 

200 

17.7 

17.5 

17.7 

17.7 

17.6)18.1 

18.3 

18.7 

19.2 

19.7 

20.1 

20.8 

21.5 

220 

18.3 

18.2 

18.3 

18.3 

18.3  |  18.3 

18.6 

18.7 

18.9 

19.3 

19.5 

20.0 

20.4 

240 

18.6 

18.8 

18.9 

18.9 

18.9 

19.0 

19.2 

19.1 

19.2 

19.5 

19.6 

19.7 

19.9 

260 

18.2 

18.5 

18.7 

18.8 

19.0 

19.3 

19.5 

19.6 

19.9 

19.9 

20.0 

20.1 

20.2 

280 

16.8 

17.4 

17.9 

18.3 

18.7 

19.1 

19.3 

19.8 

20.0 

20.2 

20.4 

20.6 

20.8 

300 

15.5 

15.8 

16.2 

16.6 

17.6 

18.1 

18.5 

19.2 

19.4 

19.9 

20.6 

20.8 

20.9 

320 

13.8 

14.2 

14.6 

15.1 

15.6 

16.2 

16.8 

17.7 

18.3 

18.9 

19.5 

20.1 

20.8 

340 

12.6 

12.9 

13.0 

13.3 

13.7 

14.4 

14.9 

15.5 

16.2 

17.1 

18.0 

18.6 

19.4 

360 

12.3 

12.1 

11.9 

12.0 

12.3 

12.5 

13.0 

13.4 

14.2 

14.9 

15.7 

16.5 

17.3 

380 

13.1 

12.5 

11.9 

11.6 

11.5 

11.4 

11.6 

11.7 

12.3 

12.7 

13.3 

14.0 

15.0 

400 

14.8 

13.9 

13.1 

12.5 

11.7 

11.2 

11.1 

10.9 

11.0 

11.1 

11.4 

12.0 

12.6 

420 

16.5 

15.7 

15.1 

14.3 

13.4 

12.5 

11.7 

11.1 

10.8 

10.8 

10.5 

10.6 

10.7 

440 

18.6 

17.9 

17.1 

16.1 

15.6 

14.4 

13.5 

12.8 

11.9 

11.1 

10.6 

10.3 

10.3 

460 

20.3 

19.8 

19.3 

18.5 

17.6 

16.8 

15.9 

14.7 

13.7 

12.9 

12.0 

11.1 

10.9 

480 

21.2 

21.1 

20.8 

20.3 

19.7 

19.1 

18.3 

17.4 

16.4 

15.0 

14.1 

13.2 

12.2 

500 

21.4 

21.4 

21.4 

21.3 

21.1 

20.8 

20.0 

19.5 

18.8 

17.8 

17.0 

15.7 

14.4 

520 

20.6 

21.2 

21.7 

21.7 

21.5 

21.5 

21.4 

21.1 

20.5 

19.8 

19.1 

18.2 

17.6 

540 

19.2 

20.0 

20.7 

21.1 

21.8 

22.0 

21.8 

21.7 

21.5 

21.2 

20.9 

20.3 

19.6 

560 

17.2 

18.4 

19.0 

20.0 

20.8 

21.1 

22.7 

21.9 

22.2 

22.1 

21.9 

21.7 

21.1 

5SO 

15.0 

16.0 

17.3 

18.2 

19.1 

19.9 

20.8 

21.1 

21.7 

22.0 

22.2 

22.3 

22.1 

600 

12.8 

13.9 

15.1 

15.9 

17.2 

18.0 

19.0 

19.9 

20.6 

21.3 

21.8 

22.0 

22.4 

620 

10.6  1  11.5 

12.7 

13.7 

14.9 

16.0 

17.1 

18.3 

19.1 

19.9 

20.8 

21.3 

22.0 

640 

8.5 

9.5 

10.4 

11.3 

12.3 

13.7 

14.9 

16.0 

17.1 

18.1 

19.0 

19.9 

20.7 

660 

6.8 

7.4 

8.2 

9.1 

10.1 

11.1 

12.2 

13.6 

14.6 

15.8 

17.1 

18.1 

19.0 

680 

5.1 

5.7 

6.4 

7.1 

7.9 

8.7 

9.7 

11.0 

12.1 

13.1 

14.1 

15.7 

16.8 

700 

4.2 

4.4 

4.7 

5.1 

5.8 

6.7 

7.4 

8.4 

9.4 

10.6 

11.5 

13.0 

14.1 

720 

3.8 

3.8 

3.8 

4.0 

4.4 

4.8 

5.4 

5.9 

6.9 

8.0 

9.1 

10.1 

11.5 

740 

4.3 

3.9 

3.8 

3.7 

3.6 

3.8 

3.9 

4.4 

4.9 

5.7 

6.4 

7.4 

8.9 

760 

5.9 

5.1 

4.4 

4.0 

3.6 

3.4 

3.4 

3.5 

3.9 

4.3 

4.7 

5.2 

5.9 

780 

8.1 

7.1 

6.1 

5.3 

4.6 

4.1 

3.7 

3.3 

3.3 

3.1 

3.4 

3.6 

4.1 

800 

10.8 

9.7 

8.5 

7.5 

6.5 

5.6 

4.9 

4.2 

3.8 

3.4 

3.2 

3.1 

3.1 

820 

13.6 

12.5 

11.2 

10.1 

9.0 

8.0 

6.9 

6.1  5.3 

4.7 

3.9 

3.7 

3.1 

840 

16.4 

15.1 

1C.  7 

12.9 

11.7 

10.6 

9.5 

8.6  7.5 

6.6 

5.7 

49 

4.4 

860 

18.4 

17.5 

16.6 

15.4 

14.3 

13.1 

12.1 

11.1  10.0 

9.1 

7.9 

7.0 

6.3 

880 

20.41  19.6 

18.7 

17.5  i  16.  6  15.6  14.5  13.6  12.5  11.5 

10.4 

9.5 

8.6  i 

900 

22.0  21.1 

20.2 

19.4 

18.7 

17.7 

16.5  15.7  14.7  13.8 

12.5 

11.9! 

109 

920 

23.6  1  22.7 

21.7 

21.1 

20.1 

19.4 

18.4  j  17.5  16.7  15.6 

14.8 

13.9 

13.1 

940 

25.5124.1 

23.4 

22.4 

21.4  20.6  il9.9  19.0  18.2  17.3 

16.6 

15.7 

14.8 

960 

26.2  25.6 

24.7 

24.1 

23.3  223  21.3  20.6  19.0  18.9 

17.9 

17.1 

16.31 

980 

27.1  26.7  1  26.3 

25.5 

24.9  23.8  23.4  22.2  21.0  20.4 

19.4 

18.6 

17.7 

1000 

27.6  27.7 

27.3 

26.7  26.2,25.5:24.7  23.8  23.1  22.3 

21.3  20.2 

19.3| 

360  370 

380 

390  400  •.  410  420  430  .  440  450 

460  470 

480 

28 


TABLE  XXX. 


Perturbations  produced  by  Venus. 

Arguments  II  and  III. 

III. 


II. 

480 

490 

500 

510 

520 

530 

|  540 

550 

560 

570 

580 

690  COO 

0 

19.3 

18.3 

17.4 

16.6 

15.7 

15.0 

14.2 

13.6 

13.1 

12.3 

11.7 

11.3  108 

20 

20.9 

20.2 

19.1 

18.2 

17.1 

16.2  15.5 

14.7 

14.1 

13.3 

12.7 

12.2  11.5 

40  '  23.2 

22.0 

20.8 

20.1  18.9 

17.9!  17.1 

15.9 

15.1 

14.4 

13.7 

13.0  12.:j 

60  |  V5.0 

24.0 

23.2 

22.0  !  20.7 

19.9;  18.9 

17.7 

16.8 

15.8 

14.9 

14.0  i  13.3 

80  27.2 

26.4 

25.6 

24.1  |23.2 

22.1  :20.8 

20.0 

18.7 

17.9 

16.6 

15.6  14.8 

100  1  27.4 

27.2 

26.8 

26.3 

25.4 

24.5  ,  23.5 

22.2 

20.9 

20.0 

18.6 

17.6 

16.6 

1  120 

27.5 

27.5 

27.6 

27.1 

26.8 

26.3  !  25.4 

24.6 

23.7 

22.4 

21.0 

20.1 

i  18.8 

140 

26.7 

27.0 

27.2  27.4 

27.3 

27.4  26.9 

26.2 

25.4 

24.6 

23.9 

22.6  '21  1 

HiO 

25.1 

25.6 

26.1  26.7 

26.9 

27.3  27.1 

27.0 

26.9 

26.4 

25.5 

24.  7  I  23.  9 

ISO 

23.0 

23.8 

24.5  i  25.0 

25.7 

26.3  26.7 

26.8 

27.0 

26.8 

26.6 

•26.2 

25.6 

200 

21.5 

22.2 

22.8  23.5 

24.1 

24.7  j  25.5 

25.8 

26.3 

26.6 

26.6 

26.6 

26.4 

220 

20.4 

21.0 

21.5 

22.0 

22.6 

23.2 

23.8 

24.5 

25.0 

25.4 

25.8 

26.0 

26.2 

240 

19.9 

20.4 

20.8 

21.2 

21.6 

21.8 

1  22.2 

22.6 

23  1 

23.3 

23.9 

24.2 

24.6 

260 

20.2 

20.3 

20.6 

21.2 

21.4 

21.7 

21.9 

22.2 

223 

22.7 

23.1 

23.3 

23.6 

280 

20.8 

20.8 

21.0 

21.1 

21.3 

21.4  21.5 

21.8 

22.0 

22.2 

22.7 

23.0 

23.3 

300 

20.9 

21.0 

21.5 

21.7 

21.7 

22.0 

22.0 

22.1 

22  1 

22.2 

224 

22.6 

22.8 

320 

20.8 

21.2 

21.5 

31.6 

22.0 

22.3 

22.5 

22.5 

226 

22.7 

22.8 

22.8 

22.9 

340 

19.4 

20.2 

20.8 

21.5 

21.9 

22.1 

22.6 

23.0 

23.2 

23.4 

23.3 

23.4 

23.5 

360 

1-7.3 

18.4 

19.5 

20.0 

20.6 

21.5 

22.2 

22.7 

23.0 

23.7 

23.7 

24.0 

24,2 

380 

15.0 

15.9 

16.9 

17.8 

18.6 

19.6 

20.6 

21.5 

22.3 

22.9 

235 

23.9 

24.5 

400 

12.6 

13.2 

14.2 

15.4 

16.2 

17.3 

18.1 

19.2 

20.3 

21.4 

224 

23.0 

23.7 

420 

10.7 

11.2 

12.0, 

12.5 

13.5 

14.5 

15.6 

16.7 

17.7 

18.7 

20  I 

21.0 

22.0 

440 

10.3 

10.2 

10.3 

10.5 

11.3 

12.0 

12.9 

13.6 

14.7 

16.0 

17.0 

18.3 

19.5 

460  10.9 

10.1 

9.9 

9.9 

9.9 

10.1 

10.7 

11.3 

12.2 

13.0 

140 

15.1 

16.5 

480 

12.2 

11.4 

10.7 

10.1 

9.7 

9.5 

9.7 

9.9 

10.2 

10.7 

11.7 

12.5 

134 

500 

14.4 

13.6 

12.5 

11.6 

10.9 

10.2 

9.8 

9.4 

9.3 

9.6 

9.8 

10.2 

11.1 

520 

17.6 

16.2 

15.1 

13.9 

12.9 

11.9 

10.9 

10.3 

9.8 

9.5 

9.2 

9.2 

9.6 

540 

19.6 

18.6 

18.0 

16.7 

15.4 

14.5 

12.2 

12.3 

11.3 

10.5 

10.1 

95 

9.3 

560 

21.1 

20.4 

19.8 

19.0 

18.2 

17.2 

16.0 

14.8 

13.7 

12.7 

11.7 

10.9 

10.2 

580 

22.1 

21.8 

21.5 

20.9 

20.3 

19.3 

18.6 

17.3 

16.5 

15.4 

14.0 

129 

12  2 

600 

22.4 

22.4 

22.2 

22.2 

21.5 

21.2 

206 

19.5 

19.1 

17.7 

16.8 

15.8 

U.I 

620 

22.0 

22.3 

22  4 

22.4 

22.3 

22.3 

21  9 

21.5 

20.9 

20.0 

19.3 

18.0 

16.9 

640 

20.7 

21.7 

22.0 

22.3 

22.6 

22.5 

226 

22.4 

22.0 

21.6 

21.1 

203 

19.6 

660 

19.0 

20.0 

20.8 

21.3 

22.1 

22.3 

226 

22.8 

22.7 

22.6 

22.2 

21.8 

21.3 

680 

16.8 

18.0 

19.0 

19.9 

20.8 

21.5  22  1 

22.6 

22.7 

23.0 

23.0 

22.8 

224 

700 

14.1 

15.2 

16.8 

17.9 

18.8 

20  0  22  1 

21.5 

22  2 

22.6 

22.9 

230 

23.5: 

720 

11.5 

12.7 

13.9 

15.0 

16.4 

17.9  18.6 

19.7 

20.8 

21.6 

23  3 

227 

23.0 

740 

8.9 

9.8 

10.9 

12.2 

13.6 

14.8 

16.2 

17.5 

18.7 

19.5 

20.6 

21.6 

22.3 

760 

.5.9 

6.8 

8.0 

9.3 

10.3 

11.8 

13.2 

14.5 

15.9 

17.4 

18.2 

19.5 

20.5 

780 

4.1 

4.9 

5.6 

6.4 

7.5 

8.6 

9.9 

11.  1 

12.6 

14.0 

15.6 

J6.8 

18.1 

800 

3.1 

3.3 

4.4 

4.8  _ 

5.5 

6.1 

6.9 

7.9 

9.4 

10.7 

12.1 

13.4 

14.9 

820 

3.1 

3.1 

3.2 

3.1 

3.6 

3.9 

4.8 

5.7 

65 

75 

8.7 

10.0 

11.5 

840 

4.4 

3.7 

3.5 

3.2 

3.2 

3.1 

3.4 

3.7 

4.1 

5.0 

6.2 

7.0 

8.2 

860 

6.3 

5.5 

4.6 

4.1 

3.6 

3.4'  3.3 

3.2 

3.4 

3.4 

4.0 

4.5 

5.6 

880 

8.6 

7.6 

6.7 

5.9 

5.2 

4.5  4.1 

3.8 

3.5 

34 

34 

3.6 

3.9 

900 

10.9 

10.0 

9.1 

8.3 

7.2 

6.5!  5.8 

5.1 

4.4 

42 

3.8 

3.6 

3.6 

920 

13.1 

12.1 

11.2 

10.3 

9.6 

8.7   7.7 

6.9 

6.3 

5.8 

5.1 

4.6 

4.2 

940 

14.8 

14.1 

13.1 

12.4 

11.5 

10.8  9.8 

9.1 

8.3 

7.6 

6.8 

6.5 

5.9 

960 

16.3 

15.4 

14.6 

14.0 

13.2 

12.6  11.7 

11.0 

10.1 

9.6  8.8 

8.1 

7.5 

980 

17.7 

16.8 

16.2 

15.2 

14.5 

13.9  13.1 

12.5 

11.8 

11.2 

10.5 

9.7 

9.3 

1000 

19.3 

18.3 

17.4 

16.6 

15.7 

15.0  ;  14.2 

13.6 

13.1 

12.3 

11.7 

11.3 

10.8 

480 

490 

500 

510 

520 

530  540 

550 

500 

570  580 

590 

600 

TABLE  XXX. 


29 


Perturbations  produced  by  Venus. 

Arguments  II.  and  III. 

1IL 


H.   600  610  620 

630  640  650  660 

670 

680 

690 

700 

710 

720 

I  "  i  "    " 

0  10.8  10.2|  9.5 

9.1  8.4 

7.9 

7.4 

7.0  6.6  6.3 

5.9 

5.5 

5.4 

20  11.5  11.3  10.7 

10.4)  9.8 

9.4 

8.9 

8.5  7.9   7.7 

7.3 

6.7 

6.6 

40  12.3  12.0  ;  11.5!  11.0  10.7  10.3  10.0 

9.6  9.3  8.9 

8.5 

8.1 

7.8 

60  13.3  12.7  12.1  i  11.6  11.2  10.9  10.5 

10.2  10.0  9.8 

9.5 

9.2 

8.9 

80  M.S  13.6  12.9;12.4  11.8  11.3  10.9 

10.7  10.3   9.9 

9.8 

9.8 

9.6 

100  16.0  15.4 

14.4 

13.4  12.6 

12.1  j  11.5 

11.0  10.6  10  2 

10.0 

9.9 

9.6 

120  |  18.S|  17.7 

16.4 

15.3  14.3 

13.21  12.4 

11.6  11.2  10.6 

10.1 

10.1 

9.6 

140  21.1 

20.1 

18.9 

17.7  16.5  15.2 

14.2 

13.0  12.3  11.6 

11.1 

10.3 

9.9 

160 

23.9 

22.9 

21.5 

20.4  19.2  17.9 

16.6 

15.3  14.1  13.1 

12.0 

11.2 

10.5 

180 

25.6 

24.8 

23.9 

22.9 

21.6 

20.6 

19.1 

18.0  16.7  15.5 

14.3 

12.9 

12.0 

200 

26.4 

26.0- 

25.6 

24.9 

24.0 

22.9 

21.7 

20.8  19.3 

18.1 

16.9 

15.5 

14.4 

220 

26.2 

26.3 

26.1 

25.8 

25.3 

24.9 

24.1 

23.1  21.2  20.9 

19.7 

18  3 

17.1 

240 

24.6 

25.1 

25.1 

25.3 

25.2 

25.1 

24.7 

24.3  24.0  23.0 

21.9 

21.3 

20.2 

260 

23.6 

23.9 

24.2 

24.5 

24.7 

24.8 

24.9 

24.6  24.3  23.8 

23.4 

22.9 

21.6 

280 

23.3 

23.6 

23~9 

24.2 

24.7 

24.8 

25.0 

24.9  24.9 

24.8 

24.4 

24.0 

23.5 

300 

22.8 

23.0 

23.3 

23.4 

23.8 

24.0 

24.1 

24.5 

24.5 

24.6 

24.5 

24.4 

24.0 

320 

22.9 

23.0 

23.1 

23.2 

23.4 

23.3 

23.6 

23.8 

24.0 

23.9 

24.2 

24.2 

24.2 

340 

23.5 

23.5 

23.5 

23.4 

23.5 

23.6 

23.6 

23.5  23.5 

23.6 

23.9 

23.8 

23.8 

360 

24.2 

24.2 

24.3 

24.2 

24.2 

24.0 

23.7 

23.9  24.0 

23.7 

23.7 

23.6 

23.6 

380 

24.5 

24.6 

24.8 

25.1 

24.8 

24.9 

25.0 

24.9  24.6 

24.5 

24.5 

24.3 

24.0 

400 

23.7 

24.3 

24.7 

25.0 

25.4 

25.7 

25.7 

25.5 

25.5 

25.4 

25.2 

24.8 

24.6 

420 

22.0 

23.0 

23.7 

24.6 

25.0 

25.7 

26.1 

26.2 

26.3 

26.5 

26.2 

26.0 

25.9 

440 

19.5 

20.8 

21.7 

22.7 

23.7 

24.6 

25.4 

26.0  26.5 

26.7 

26.9 

27.0 

26.9 

460 

16.5 

17.8 

19.0 

20.1 

21.4 

22.3 

23.5 

24.8  i  25.4 

26.1 

26.7 

27.1 

27.3 

4SO 

13.4 

14.5 

15.6 

17.0 

18.5 

19.7 

20.9 

22.1  i  23.2  24.4 

25.4 

26.2 

26.8 

500 

11.1 

12.0 

13.0 

13.8 

14.9 

16.3 

17.9 

19.1 

20.5 

21.6 

22.9 

24.2 

25.1 

520 

9.6 

9.8 

10.5 

11.5 

12.4 

13.4 

14.4 

15.5 

17.1 

18.4 

19.9 

21.2 

22.3 

540 

9.3 

9.0 

9.2 

9.6 

10.3 

11.0 

11.9 

12.8 

13.9 

15.1 

16.5 

17.9 

19.4 

560 

10.2 

9.7 

9.3 

9.1 

9.1 

9.4 

10.0 

10.6 

11.5 

12.4 

13.3 

14.5 

16.0 

580 

12.2 

11.3 

10.4 

9.9 

9.4 

9.0 

9.2 

9.3 

9.7 

10.4 

11.0 

12.0 

12.7 

600 

14.4 

13.3 

12.5 

11.6 

10.8 

10.1 

9.6 

9.4 

9.1 

9.3 

9.9 

10.0 

10.8 

620 

16.9 

16.1 

14.9 

13.7 

12.7 

12.0 

11.1 

10.4 

9.8 

9.5 

9.5 

9.3 

9.7 

640 

19.6 

18.4 

17.4 

16.3 

15.2 

14.2 

13.1 

12.1 

11.3 

10.6 

10.1 

9.6 

9.5 

660 

21.3  20.6  1  19.9 

18.7 

17.8 

16.7 

15.6 

14.4 

13.4 

12.4 

11.7 

11.0 

10.2 

,680 

224  22.0  I  21.  5  20.8 

20.2 

19.0 

18.1 

17.0 

15.8  i  14.7 

13.7 

12.8 

12.0 

700 

23.2  23.2  |  22.6  22.2 

21.7 

21.0 

20.5 

19.3 

18.3  |  17.3 

16.0 

15.0 

14.1 

720 

23.0  23  3  23.2  23  4 

23.1 

22.4 

21.9 

21.3 

20.8  1  19.5 

18.5 

17.6 

16.4 

740 

22.3  22.8  23.2  23.4  23.6 

23.6  23.3 

22.8 

22.2i21.6 

21.1 

19.9 

18.8 

760 

20.5  21.4  22.5  22.8  23.3 

23.7  23.6 

23.8  23.5  23.3 

22.7 

21.8 

21.3 

780  18.1  19.2  20.4  21.3  22.3  23.0  23.3 

23.7 

23.8  24.0 

23.8 

23.5 

23.0 

800 

14.9  16.4  i  17.7  1  19.1  ,  20.1  21.2  j  21.1 

22.9 

23.4  23.8 

24.1 

24.2 

23.9 

820 

11.5  12.9  14.3  15.8  17.8  '  18.7  20.0 

20.9  22.0  22.7 

23.5 

23.9 

24.0 

840 

82  9.5  10.8  12.2  13.8  15.2  16.6 

18.1  19.5  20.6 

21.7 

22.6 

23.3 

860 

5.6   6.8   7.7  8.8  10.2  11.5  j  13.2 

14.7 

16.0  17.4 

19.0 

20.2 

21.3 

880 

39  4.4 

5.2 

6.1 

7.2 

8.2  I  9.7 

10.9 

12.5  14.1 

15.4 

16.8 

18.2 

900 

3.6 

3.6 

3.9 

4.2 

5.0 

5.7 

6.6 

7.8 

9.1  10.3 

11.8 

13.4 

14.8 

920 

4.2 

3.8 

3.9 

3.9 

4.0 

4.3 

4.7 

5.4 

6.4   7.3 

8.6 

9.8 

11.2 

940   5.9  5.1 

4.6  4.4 

4.2 

4.3  4.3 

4.3 

4.9  5.3 

6.3 

7.0 

8.0 

960   7.5   6.9 

6.3 

5.8 

5.3 

4.7  4.7 

4.6 

4.6  4.6 

4.9 

5.4 

6.0 

980   9.3   8.7 

7.9 

7.4 

6.8 

6.4  6.0 

5.6 

5.2  5.0 

4.9 

5.1   5.1 

'1000  i  10.  8  10.2 

9.5 

9.1 

8.4 

7.9 

7.4 

7.0  6.6  6.3 

5.9 

5.5  5.4 

600  610  620  i  630  i  640  650  660  670  ;  680  690  [  700  j 710  ;  720 


30 


TABLE  XXX. 


Perturbations  produced  by   Venus. 
Arguments  II.  and  III. 

in. 


IL   720 

730 

740 

750 

760 

770 

780 

790  800 

810 

820  830 

840 

0!  5.4 

// 
5.5 

!  5.8 

6.0 

6.3 

6.8 

7.6 

8.4 

9.3 

10.4 

11.7  12.9 

14.3 

20   6.6 

6.3 

6.0 

6.1 

6.1 

6.2 

6.5 

6.9 

7.7 

8.3 

9.4 

10.2 

11.2 

40   7.8 

7.4 

:  7.1 

7.0 

6.7 

6.6 

6.8 

6.8 

6.9 

7.2 

7.7 

8.5 

9.3 

60   8.9 

8.8 

'  8.3 

8.1 

7.8 

7.6 

.7.4 

7.4 

7.3 

7.4 

7.4 

7.7 

8.3 

80  ;  9.6 

9.5 

:  9.1 

9.1 

9.0 

8.8 

8.4 

8.2 

8.1 

8.1 

8.0 

8.1 

8.2 

100!  9.6 

9.5 

j  9.6 

9.5 

9.5 

9.3 

9.3 

9.2 

9.2 

9.0 

8.7 

8.7 

8.7 

120  9.6 

9.6 

1  9.5 

9.3 

9.4  i  9.6 

9.6 

9.5 

9.5 

9.6 

9.6 

9.6 

9.6 

140  9.9 

9.5 

|  9.6 

9.4 

9.3  9.3 

9.0 

9.3 

9.5 

9.8 

9.7 

9.8 

10.0 

i60  10.5 

9.9 

1  9.5 

9.1 

8.9 

9.0 

8.9 

9.0 

9.0 

9.0 

9.5 

9.6 

9.9 

180  12.0 

11.0 

10.1 

9.7 

9.1 

8.8 

8.7 

8.3 

8.5 

8.7 

8.8 

9.0 

9.1 

200  14.4 

13.3 

12.0 

11.0 

10.1 

9.4 

8.9 

8.5 

8.2 

8.0 

8.0 

8.3 

8.5 

220  17.1 

15.7 

'l4.6 

13.2 

12.0 

10.9 

10.2 

9.2 

8.7 

8.3 

7.9 

7.7 

7.7 

240  20.2 

19.1 

i  17.8 

16.5 

14.5 

13.4 

12.2 

11.1 

10.0 

9.4 

8.4 

8.0 

7.7 

260  21.6 

21.1 

!20.1 

19.2 

17.3 

15.9 

14.6 

13.4 

12.4 

11.3 

10-1 

9.1 

8.6 

280  23.5 

22.7 

,21.6 

21.0 

19.8 

18.8 

17.3 

16.1 

15.0 

13.5 

12.5 

11.5 

10.2 

300  24.0 

23.4 

23.2 

22.4121.4 

20.5 

19.8 

18.7 

17.5 

16.1 

15.0 

13.7 

12.4 

320  24.2 

23.9 

23.5 

23.1 

22.7 

22.2 

21.2 

20.6 

19.6 

18.6 

17.5 

16.3 

15.1 

340  23.8 

23.9 

23.7 

23.5 

23.2 

22.8 

22.3 

21.4 

20.9 

20.5 

19.2 

18.6 

17.4 

360  23.6 

23.6 

23.6 

23.3 

23.3 

23.1 

22.9 

22.4 

22.0 

21.4 

20.4 

19.9 

18.9 

380  i  24.0 

24.0 

23.7 

23.5 

23.3 

23.1 

23.1 

22.7 

22.4 

22.2 

21.6 

20.8 

20.0 

400  24.6 

24.4 

24.4 

24.0 

23.8 

23.4 

23.2 

23.0 

22.8 

22^4 

22.1 

21.6 

21.3 

420  25.9 

25.6 

25.2 

24.8 

24.7 

24.3 

23.9 

23.6 

23.3 

22.9 

22.7 

22.3 

21.7 

440  1  26.9 

26.6 

26.4 

20.2 

25.9 

25.5 

25.2 

24.9 

24.5 

238 

23.4 

23.0 

22.8 

460 

27.3 

27.6 

27.6 

27.4 

27.0 

26.9 

26.5 

26.1 

25.6 

25.0 

24.6 

24.2 

23.7 

480 

26.8 

27.4 

27.6 

28.0 

28.1 

28.2 

27.7 

27.4 

27.3 

26.6 

26.2 

25.7 

25.1 

500 

25.1 

26.1 

26.8 

27.5 

28.1 

28.2 

28.6 

28.5 

28.4 

28.3 

27.6 

27.2 

26.7 

520 

22.3  23.9 

24.8 

25.9 

26.8 

27.5 

28.1 

28.5 

28.7 

29.0 

28.8 

28.6 

28.4 

540 

19.4 

20.7 

22.1 

23.4 

24.6 

25.6 

26.5 

27.4 

28.0 

28.7 

28.9 

29.1 

29.2 

560 

16.0 

17.3 

18.6 

19.9 

21.4 

22.9 

24.1 

25.5 

26.4 

27.3 

28.2 

28.6 

29.2 

580 

12.7 

14.1 

15.5 

16.8 

18.0 

19.3 

20.9 

22.2 

23.5 

24.9 

26.1 

27.0 

27.8 

600 

10.8 

11.6 

12.7 

13.6 

14.9 

16.2 

17.5 

18.7 

20.2 

21.8 

23.0 

24.4 

25.5 

620 

97 

10.0 

10.5 

10.7 

12.2 

13.2 

14.4 

15.6 

17.0 

18.3 

19.6 

21.2 

22.6 

640 

9.5 

9.4 

9.6 

10.1 

10.4 

11.1 

12.0 

13.0 

14.0 

15.2 

16.5 

17.9 

19.2 

660 

10.2 

10.0 

9.7 

9.5 

9.5 

9.9 

10.4 

11.0 

11.7 

12.7 

13.8 

14.9 

16.2 

680 

12.0 

11.2 

10.5 

10.0 

9.7 

9.5 

9.6 

10.0 

10.4 

11.0 

11.6 

12.5 

13.8 

700 

14.1 

13.1 

12.3 

11.3 

10.7 

10.1 

9.7 

9.7 

9.9 

9.9 

10.4 

10.9 

11.5 

720 

16.4 

15.3 

14.4 

13.3 

12.2 

11.6 

10.9 

10.2 

10.1 

9.9 

10.0 

10.1 

10.4 

740 

18.8|  17.7 

16.7 

15.6 

14.4 

13.5 

12.4  11.5 

11.1 

10.7 

10.1 

10.0 

10.3 

760 

21.3  20.1 

19.2 

18.1 

16.6 

15.6 

14.7  13.6 

12.8 

11.9 

11.3 

10.7 

10.3 

780 

23.0  22.3 

21.5 

20.5 

19.4 

18.4 

17.2 

15.8 

14.9 

14.0 

13.0 

12.2 

11.3 

800 

23.9  23.9 

23.4 

22.6  j  21.  9 

20.7 

19.8 

18.8 

17.5 

16.2 

15.1 

14.2 

13.4 

820 

24.0 

24.5 

24.2 

23.9 

23.3 

22.6 

22.3 

21.3 

20.3 

19.4 

18.3 

17.3 

16.2 

|  840 

233 

24.0 

24:3 

24.5 

24.4 

24.3 

23.8 

23.4 

22.7 

21.7 

20.8 

19.6 

18.3 

860 

21.3 

22.3 

23.3 

23.9 

24.2 

24.7 

24.5 

24.5 

24.3 

23.6 

23.1 

21.9 

21.0 

1  880 

18.2 

19.7 

20.9 

22.0 

22.8 

23.8 

24.1 

24.6 

24.8 

24.7 

24.5 

24.0 

23.5 

900 

148 

16.1 

17.6 

19.0 

20.6 

21.5 

22.5 

23.2 

24.1 

24.5 

24.2 

24.8  24.5 

920 

11.2 

12.6 

14.0 

15.5 

17.0 

18.4 

19.9 

21.0 

22.0 

22.9 

23.5 

24.5 

24.5 

•  940 

8.0 

9.3 

10.7 

12.0 

13.3 

14  8 

16.4 

17.6 

19.1 

20.4 

21.4 

22.4  23.2 

960 

6.0 

6.9 

7.8 

8.6 

10.2  11.5 

12.7 

14.1 

15.6 

16.9 

18.5 

19.5  20.7 

980 

5.1 

5.5 

6.0 

6.7 

7.7 

8.5 

C.7 

10.9 

12.2 

13.6 

14.8 

16.1 

17.6 

1000 

5.4 

5.5 

5.8 

5.8 

6.3 

6.8 

7.6 

8.4 

9.3 

10.5 

11.7 

12.9 

14.3 

720 

730 

740 

750 

760  770 

780 

790 

800 

810  820 

830  840  .'- 
j 

TABLE  XXX. 


31 


Perturbations  produced  by   Venus. 
Arguments  II.  and  III. 

m. 


II.   840  •  850 

I 

860  870  880  .  890 

900  910  i  920 

930  940 

950 

960! 

1  t 
0  !  14.3 

i 

—  —  ^^—  - 

~~~~~ 

15.5 

16.9 

18.2  19.2  20.2 

21.4  22.5  23.0 

23.5 

24.0 

24.2 

24.2 

20 

ii.  a 

12.4 

13.6 

14.9  16.2  17.3 

18.6  ;  19.6  20.5 

21.5  22.4 

23.1 

23.6 

40 

9.3 

10.2 

10.9  11.8  13.3  14.2 

15.5  16.6  17.8 

18.8  |  19.7 

20.7 

21.6 

60 

8.3 

8.7 

9.5 

10.1  10.8  ill.  6 

12.7 

13.8  14.9 

15.9]  17.0 

18.1 

19.1 

80 

8.2 

8.3 

8.6 

8.9 

9.6 

10.3 

10.7 

11.6 

12.5 

13.3  14.5 

15.2 

16.2 

100 

8.7 

8.7 

8.9 

9.0 

9.1 

9.4 

9.9 

10.4 

11.0 

11.7 

12.4 

129 

14.0 

120 

96 

9.5 

9.3 

9.6 

9.6 

9.7 

99 

9.8 

10.4 

10.9 

11.3 

11.8 

12.3 

140 

10.0 

10.2 

10.1 

10.2  10.1 

10.3 

10.4 

10.5 

10.5 

i0.6 

10.9 

11.4 

11.5 

160 

9.9 

10.0 

10.2 

10.4  10.6 

11.0 

11.0 

10.9 

11.0 

11.3 

11.3 

11.3 

11.6 

180 

9.1 

9.6 

9.9 

10.1 

10.4 

10.7 

11.0 

11.3 

11.5 

11.7 

11.7 

11.9 

12.2 

200 

8.5 

8.8 

9.1 

9.5 

9.7 

10.0 

10.5 

11.0 

11.2 

11.6 

12.0 

12.2 

12.4 

220 

7.7 

7.7 

8.1 

8.4 

8.8 

9.2 

9.7 

10.1 

10.6 

11.0 

11.4 

11.8 

12.3 

240 

7.7 

7.3 

7.4 

7.4 

7.7 

8.0 

8.4 

9.0 

9.6 

10.0 

10.5 

11.0 

11.5 

260 

8.6  1  7.9 

7.4 

7.2 

7.1 

7.1 

7.3 

7.6 

8.1 

8.5 

9.3 

LO.O 

10.4 

280 

10.2 

9.2 

8.3 

7.9 

7.4 

7.1 

7.0 

6.9 

7.0 

7.3 

7.7 

8.5 

8.8 

300 

12.4 

11.4 

10.4 

9.3 

8.5 

7.8 

7.4 

6.9 

6.7 

6.8 

6.8 

7.0 

7.5 

320 

15.1 

13.9 

12.5 

11.4 

10.5 

9.7 

8.6 

7.8 

7.4 

7.0 

6.6 

6.5 

6.7 

340 

17.4 

16.4 

15.2 

13.9 

12.7 

11.6 

10.6 

9.7 

8.7 

8.0 

7.3 

6.8 

6.6 

360 

18.9 

18.1 

174 

16.3 

15.1 

13.8 

12.8 

11.7 

10.6 

9.8 

8.8 

8.0 

7.4 

380 

20.0 

19.6 

18.8 

17.7 

16.9 

13.0 

15.1 

13.9 

12.7 

11.8 

10.8 

9.8 

8.9 

400 

21.3 

20.6 

19.6 

19.4 

18.4 

17.6 

16.5 

15.7 

14.8 

13.7 

12.8 

11.8 

10.9 

420 

21.7 

21.1 

20.8 

20.3 

19.3 

18.9 

18.2 

17.2 

16.3 

15.3 

14.5 

13.7 

12.6 

440 

22.8 

22.1 

21.6 

20.8 

20.6 

19.7 

19.0 

18.6 

17.7 

16.6 

15.9 

15.1 

14.2 

460 

23.7 

23.3 

22.7 

22.0 

21.6 

20.9 

20.2 

19.5 

18.5 

18.1 

17.3 

16.7 

15.7 

180 

25.1124.4 

23.9 

23.3 

22.8 

22.0 

21.4 

20.9 

20.2 

19.3 

18.3 

17.7 

16.9 

500 

26.7126.3 

25.7 

24.9 

24.3 

23.6 

23.0 

22.3 

21.4 

20.7 

20.3 

19.1 

18.1 

520 

28.4127.8 

27.3 

26.8 

26.3 

25.6 

24.7 

23.9 

23.3 

22.6 

21.8 

20.8 

20.1 

540 

29.2  j  29.  2 

28.9 

28.5 

27.8 

27.4 

26.8 

26.1 

25.3 

24.4 

23.7 

23.0 

22.0 

560 

29.2  293 

29.5 

29.6 

29.3 

29.1 

28.8 

28.0 

27.4 

26.9 

26.1 

25.1 

24.3 

580 

27.8i28.6 

29.0 

29.4 

29.6 

29.8 

29.8 

29.3 

28.0 

28.7 

27.9 

27.3 

26.6 

600 

25.Gi26.7 

27.6 

28.4 

28.9 

29.2 

29.6 

29.9 

29.9 

29.8 

29.3 

29.0 

28.5 

620 

22.6  23.8 

25.0 

26.2127.1  27.9 

28.8 

29.3 

29.6 

29.8 

30.1 

29.8 

29.6 

640 

19.2  20.6 

21.6 

23.3 

24.6  j  25.2 

26.6 

27.8 

28.3 

28.9 

29.4 

29.7 

29.9 

660 

16.2  17.5 

18.8 

20.2 

21.1  22.9 

24.0 

25.1 

26.2 

27.1 

28.2 

28.8 

29.2 

680 

13.8,  14.7 

15.8 

16.9 

18.4  19.9 

20.6 

223 

23.6 

24.9 

25.8 

26.7 

27.5 

700 

11.5  i  12.3 

13.4 

14.6 

15.6  j  16.7 

18.0)19.5 

20.7 

22.0 

23.1 

24.2 

25.1 

720 

10.4  j  11.0 

11.4 

12.3 

13.3  '  14.3 

15.6  j  16.4 

17.7 

19.3 

19.9 

21.6 

22.6 

710 

10.3  10.4 

10.5 

11.0 

11.4!  12.2 

13.3 

14.2 

15.3 

16.5 

17.4 

18.8 

19.5 

760 

10.31  10.0 

10.2 

10.3 

10.7  :  11.0 

11.5 

12.2 

13.1 

14.2 

15.1 

16.0 

17.3 

780 

11.3 

10.8 

10.6 

10.2 

10.2  10.5 

10.7 

11.1 

11.5 

12.3 

13.2 

14.0 

15.0 

800 

13.4 

12.5 

11.7 

11.0 

10.6  10.3 

10.3 

10.4 

10.7 

11.0 

11.6 

11.3 

12.2 

820 

16.2 

15.2 

14.4 

13.5 

13.5  11.9 

11.4 

11.0 

10.9 

10.8 

10.8 

11.2 

11.4 

840 

18.3 

17.1 

16.2 

14.9 

14.1  13.0 

12.4 

11.7 

11.2 

10.7 

10.6 

11.1 

11.2 

860 

21.0 

20.2 

18.7 

17.7 

16.6  15.4 

14.3 

13.3 

12.5 

11.9 

11.4 

11.0 

10.9 

880  23.5 

22.4 

21.3 

20.4 

19.3  18.0 

17.0 

15.9 

14.8 

13.7 

12.8 

12.0 

12.6 

900 

24.5  24.2 

23.8  22.7  21.9  1Q.9 

19.7 

18.6  17.2 

16.4 

15.3 

14.1 

13.3 

920 

24.5  !  24.8 

24.7  24  3;  24.1  23.2 

22.3 

21.3  20.0 

19.3 

18.0 

16.7 

15.7 

940  23.2  24.0  24.5  24.6  j  24.5  24.5 

24.2 

23.5  22.7  21.8 

20.6  19.5 

18.4 

960  20  7  21.9  228  23.6  24.0  24.5 

24.5  24.2  i  24.3 

23.7 

22.9  22.1  21.0 

980  17.6  18.7  20.1  21.2  22.2  23.1  23.6 

24.0  i  24.3 

24.3  i  24.3  ;  23.7  23.0 

1000  14.3  15.5  16.9  18.2 

19.2  20.2;21.4 
1 

22.5  23.0 

23.5  24.0  i  24.2  24.2  j 

860  870  880  890  ;  900  i  910  |  920  !  930  j  940  i  950  |  960 


32 


TABLE  XXX.     XXXI. 


Perturbations  by   Venus. 

Arguments  II  and  III. 

III. 


Perturbations  by  Mars. 

Arguments  II  and  IV. 

IV. 


11. 

960  i  970 

980  990 

1000 

0   10 

20  j  30 

40   50 

60 

70 

0 

24.2 

23.7 

23.1 

22.5 

21.6 

9.5 

10.2 

10.8  11.2 

11.5 

11.7 

11.8 

11.5 

20 

23.6  23.7 

24.0 

23.4 

23.1 

8.3 

9.1 

9.8  10.5 

10.9 

11.2 

11.5 

11.6 

40 

21.6  22.4 

22.9 

23.5 

23.5 

7.1 

7.9 

8.8   9.4 

10.0 

10.6 

10.8 

11.2 

60 

19.1  20.1 

20.7  21.5 

22.2 

5.8 

6.7 

7.6  ;  8.4 

9.1 

9.8 

10.3 

10.5 

80 

16.2!  17.3 

18.4  19.7 

20.0 

4.3 

5.3 

6.4 

7.2 

8.0 

8.9 

9.3 

9.9 

100 

14.0  14.8 

15.6 

16.5 

17.6 

3.3 

4.2 

5.0 

5.9 

6.8 

7.6 

8.4 

9.1 

120 

12.3  |  12.9 

13.7 

14.3 

15.3 

2.4 

3.1 

3.9 

4.8 

5.6 

6.4 

7.3 

8.0 

140 

11.5  12.0 

12.6 

12.8 

13.6 

2.1 

2.4 

2.9 

3.8 

4.6 

5.5 

6.3 

7.0 

160 

11.6 

11.8 

12.1 

12.3 

12.7 

2.0 

2.2 

2.4 

2.7 

3.5 

4.4 

5.1 

5.9 

180 

12.2 

12.2 

12.3 

12.5 

12.7 

1.9 

2.0 

2.3 

2.6 

2.9 

3.4 

3.9 

4.9 

200 

12.4  i  1'2.7 

12.8 

13.1 

13.2 

2.3 

2.2 

2.2 

2.4 

2.7 

3.0 

3.4 

3.8 

220 

12.3 

12.7 

13.0 

13.3 

13.5 

3.0 

2.6 

2.5 

2.4 

2.5 

2.7 

3.1 

3.5 

240 

11.5 

12.1 

12.4 

13.1 

13.6 

3.7 

3.3 

3.0 

2.9 

2.7 

2.8 

2.9 

3.2 

260 

10.4 

11.0 

11.5 

12.2 

12.8 

4.8 

4.1 

3.7 

3.5 

3.1 

3.1 

3.0 

3.1 

280 

8.8 

9.6 

10.4 

10.7 

11.5 

5.5 

5.1 

4.6 

4.1 

3.8 

3.5 

3.5 

3.4 

300 

7.5 

7.9 

8.6 

9.0 

10.1 

6.2 

5.8 

5.6 

5.0 

4.8 

4.2 

3.9 

3.8 

.320 

6.7 

6.8 

7.3 

7.8 

8.3 

6.9 

6.6 

6.1 

5.9 

5.4 

5.1 

4.7 

4.3 

340 

6.6 

6.4 

6.6 

6.7 

6.2 

7.2 

7.1 

6.9 

6.5 

6.2 

5.8 

5.5 

5.1 

360 

7.4 

6.9 

6.5 

6.5 

6.5 

7.5 

7.4 

7.1 

7.0 

6.8 

6.4 

6.2 

5.8 

380 

8.9 

8.2 

7.5 

6.9 

6.8 

7.5 

7.6 

7.3 

7.3 

7.2 

7.1 

6.7 

6.5 

400 

10.9 

10.0 

9.0 

8.3 

7.5 

7.3 

7.3 

7.5 

7.4 

7.4 

7.4 

7.1 

7.0 

420 

12.6 

11.6 

10.7 

9.9 

9.1 

6.9 

7.0 

7.3 

7.4 

7.4 

7.4 

7.3 

7.5 

410 

14.2 

13.3 

12.5 

11.6 

10.6 

6.5 

6.8 

6.8 

7.1 

7.2 

7.3 

7.3 

7.4 

460 

15.7 

14.8 

13.9 

13.0 

12.1 

6.2 

6.2 

6.5 

6.7 

6.8 

7.1 

7.1 

7.3 

480 

16.9 

16.3 

15.5 

14.5 

13.6 

5.8 

5.9 

6.0 

6.2 

6.4 

6.5 

7.0 

6.9 

500 

18.1 

17.6 

16.6 

15.8 

15.1 

5.3 

5.4 

5.7 

5.8 

6.0 

6.0 

6.3 

6.6 

521) 

20.1 

19.2 

18.1 

17.4 

16.5 

5.1 

5.1 

5.1 

5.3 

5.4 

5.6 

5.8 

6.0 

540 

22.0 

21.0 

20.2 

19.2 

18.1 

4.7 

4.8 

4.8 

4.8 

5.0 

5.1 

5.4 

5.5 

560 

24.3 

23.5 

22.6 

21.5 

20.6 

4.4 

4.5 

4.6 

4.6 

4.7 

4.8 

4.8 

5.0 

5SO 

26.6 

25.7 

24.9 

23.8 

23.0 

4.2 

4.3 

4.4 

4.3 

4.5 

4.4 

4.4 

4.5 

600 

28.5 

27.8 

27.0 

26.3 

25.4 

4.0 

4.2 

4.3 

4.2 

4.2 

4.2 

1.2 

4.3 

620 

29.6 

29.2 

28.8 

28.2 

27.4 

4.2 

4.0 

4.1 

4.0 

4.0 

4.0 

4.0 

3.9 

640 

29.9 

30.0 

29.9 

29.5 

29.5 

4.3 

4.2 

4.1 

4.0 

4.1 

4.0 

3.9 

3.9 

660 

29.2 

29.5 

29.7 

29.8 

29.9 

4.6 

4.4 

4.3 

4.1 

4.1 

4.1 

4.0 

3.8 

680 

27.5 

28.6 

28.9 

29.2 

29.7 

4.8 

4.6 

4.5 

4.3 

4.2 

4.1 

4.0 

3.9 

700 

25.1 

26.4 

27.3 

27.8 

28.7 

5.3 

5.0 

4.8 

4.5 

4.6 

4.0 

4.1 

4.1 

720 

22.6 

23.9  25.0 

26.1 

26.8 

5.8 

5.5 

5.1 

5.0 

4.7 

4.5 

4.1 

4.1 

740 

19.5 

21.3 

22.5 

23.6 

24,6 

6.5 

6.1 

5.7 

5.4 

5.2 

4.9 

4.6 

4.3 

760 

17.3 

18.6 

19.4 

21.0 

22.1 

7.4 

6.7 

6.4  i  6.0 

5.6 

5.3 

5.1 

5.0 

780 

15.0 

15.8  17.1 

18.5 

19.3 

8.2 

7.6 

6.9  6.5 

6.4 

5.8 

5.6 

5.3 

800 

12.2 

14.1 

14.8 

15.9 

17.0 

9.2 

8.5 

8.0_  7.3 

6.8 

6.5 

6.1 

5.8 

820 

11.4 

12.0 

12.5 

13.4 

15.4 

10.1 

9.6 

8.8  '  8.2 

7.6 

7.1 

6.7 

6.5 

840 

11.2 

11.3  11.7 

12.2  13.2 

10.9 

10.4 

9.8  9.1 

8.4 

7.9 

7.5 

6.9 

860 

10.9 

10.8  !  10.9 

11.2 

11.5 

11.7 

11.0 

10.4  10.0  9.4 

8.7 

8.2 

7.7 

!  8MO 

12.6 

11.3  11.1  10.8 

11.0 

12.3 

11.9 

11.3  10.6  10.2 

9.7 

8.9 

8.4 

i  900 

13.3  12.3  12.9  11.3 

11.2 

12.4 

12.2 

11.8  11.6:10.8 

10.3 

9.7 

9.3  1 

920 

15.7  i  14.6!  13.7  12.8 

12.1 

12.3 

12.3 

12.2  11.9  !  11.6 

11.0  10.5  9.9 

940!  18.4  i  17.3  16.2  14.5 

14.0  12.1 

12.1 

12.2  12.2  11.8 

11.4  11.0  10.6 

980  121.0 

20.0  18  9  17.9 

16.7 

11.4 

11.9 

11.9  12.0  12.0 

11.7  11.4  11.0 

i  980  i  23.0 

22.4  21.4120.3 

19.5 

10.6 

11.1 

11.6  11.8  11.9 

11.9 

11.7;  11.4 

1000  i  24,2 

23.  7  |  23.1 

22.5  j  21.6 

9.5 

10.2 

10.8  11.2  11.5 

11.7 

11.8;  11.5 

[900 

970 

980 

990  1000 

0 

10 

20  ,  30   40 

50 

60 

70 

TABLE  XXXI. 

Perturbations  produced  by  Mars 

Arguments  II  and  IV. 

IV. 


33 


II. 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200, 

!  , 

0 

11.5 

11.2 

11.0 

10.6 

10.1 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

7.8 

7.6 

7.4 

20 

11.6 

11.4 

11.0 

10.9 

10.6 

10.2 

9.7 

9.1 

9.1 

8.8 

8.4 

8.1 

7.9 

7.8 

40 

11.2 

11.3 

11.2 

11.0 

10.8 

10.5 

10.3 

9.8 

9.4 

9.3 

9.1 

8.7 

8.4 

8.2 

60 

10.5 

10.9 

11.1 

10.9 

11.0 

10.9 

10.4 

10.0 

9.7 

9.5 

9.2 

8.8 

8.7 

8.4 

80 

9.9 

10.0 

10.5 

10.9 

10.8 

10.7 

10.4 

10.3 

10.0 

9.7 

9.3 

9.0 

8.8 

8.6 

100 

9.1 

9.5 

9.8 

10.1 

10.6 

10.5 

10.4 

10.3 

10.1 

9.9 

9.6 

9.3 

9.0 

8.8 

120 

8.0 

8.8 

9.3 

9.5 

9.9 

10.2 

10.2 

10.1 

10.0 

9.8 

9.6 

9.4 

9.1 

8.9 

140 

7.0 

7.9 

8.4 

9.0 

9.3 

9.6 

9.9 

99 

9.9 

9.7 

9.7 

94 

9.3 

8.9 

160 

5.9 

65 

7.2 

8.0 

8.5 

8.9 

9.2 

9.6 

9.5 

9.6 

9.5 

9.5 

9.3 

9.1 

180 

4.9 

5.6 

6^4 

6.9 

7.7 

8.3 

8.6 

8.9 

9.4 

9.3 

9.3 

9.3 

9.2 

9.1 

200 

3.8 

4.6 

5.3 

6.0 

6.7 

7.4 

7.9 

8.3 

8.0 

8.9 

9.1 

9.0 

9.0 

8.9 

220 

3.5 

39 

4.4 

5.1 

5.8 

6.4 

7.1 

7.6 

7.9 

8.4 

8.6 

8.8 

8.8 

8.7! 

240 

3.2 

3.6 

4.0 

4.4 

5.0 

5.5 

6.2 

6.8 

7.4 

7.6 

8.1 

8.4 

8.4 

8.5  i 

260 

3.1 

3.2 

3.8 

4.1 

4.5 

4.9 

5.4 

5.9 

6.6 

7.1 

7.5 

7.7 

S.O 

8.2 

280 

3.4 

34 

3.5 

3.8 

4.2 

4.5 

4.9 

5.5 

5.6 

6.2 

6.8 

7.1 

7.5 

7.8 

300 

3.S 

3.7 

3.7 

3.7 

3.9 

4.4 

4.7 

4.9 

5.4 

5.7 

6.0 

6.6 

6.9 

7.3 

320 

4.3 

4.2 

4.1 

4.0 

4.1 

4.2 

4.4 

4.7 

5.0 

5.4 

5.8 

6.0 

6.4 

6.6 

340 

5.1 

4.9 

4.6 

4.4 

4.4 

4.3 

4.5 

4.5 

5.0 

5.2 

5.5 

5.8 

6.0 

6.3 

360 

5.8 

5.6 

5.3 

5.0 

4.8 

4.8 

4.7 

4.8 

4.9 

5.1 

5.4 

5.5 

5.9 

6.1 

380 

6.5 

6.4 

5.9 

5.7 

5.5 

5.4 

5.1 

5.1 

5.1 

5.1 

5.4 

5.5 

5.7 

5.8 

400 

7.0 

6.7 

6.7 

6.3 

6.1 

5.9 

57 

5.6 

5.5 

5.5 

5.5 

5.6 

5.7 

5.9 

420 

7.4 

7.2 

6.9 

7.1 

6.7 

6.4 

6.3 

6.1 

6.0 

5.9 

5.9 

5.8 

5.8 

6.1 

440 

7.5 

7.4 

7.4 

7.0 

7.1 

7.4 

6.8 

6.7 

6.5 

6.3 

6.3 

6.4 

6.2 

6.3 

460 

7.3 

7.4 

7.4 

7.5 

7.4 

7.3 

7.3 

7.2 

7.1 

7.1 

6.7 

6.7 

6.7 

6.7 

480 

6.9 

7.1 

7.3 

7.4 

7.5 

7.3 

7.6 

7.5 

7.4 

7.5 

7.4 

7.2 

7.1 

7.1 

500 

6.6 

6.8 

6.9 

7.2 

.7.3 

7.5 

7.5 

7.6 

7.8 

7.7 

7.8 

7.7 

7.6 

7.4 

520 

6.0 

6.3 

6.5 

6.7 

7.1 

7.2 

7.5 

7.5 

7.7 

7.8 

7.9 

7.6 

7.9 

7.9 

540 

5.5 

5.7 

6.0 

6.3 

6.6 

6.9 

7.1 

7.3 

7.4 

7.7 

7.9 

8.0 

8.2 

8.3 

560 

5.0 

5.2 

5.4 

5.8 

5.9 

6.2 

6.6 

6.9 

7.1 

7.4 

7.7 

7.8 

8.1 

8.2 

580 

4.5 

4.7 

4.9 

5.0 

5.3 

5.7 

6.0 

6.6 

6.8 

7.1 

7.2 

7.5 

7.9 

8.2 

600 

4.3 

4.3 

4.4 

4.6 

4.6 

5.0 

5.3 

5.6 

5.9 

6.5 

6.9 

7.0 

7.4 

7.7 

620 

3.9 

4.0 

4.0 

4.1 

4.3 

4.4 

46 

4.9 

5.3 

5.4 

6.1 

6.6 

6.9 

7.4 

640 

3.9 

3.8 

3.8 

3.8 

3.9 

3.9 

4.1 

4.3 

4.5 

5.0 

5.2 

5.8 

6.3 

6.7 

660 

3.8 

3.7 

3.7 

3.6 

3.6 

3.7 

3.8 

3.9 

4.1 

4.2 

4.5 

5.0 

5.3 

6.0 

680 

3.9 

3.8 

3.6 

3.4 

3.5 

3.4 

3.5 

35 

3.6 

3.7 

3.8 

4.2 

4.6 

4.9 

700 

4.1 

3.9 

3.8 

3.6 

3.5 

3.3 

3.3 

3.2 

3.2 

3.2 

3.5 

3.6 

3.8 

4.2 

L. 

720 

4.1 

4.1 

4.0 

3.8 

3.6 

3.5 

3.3 

3.2 

3.3 

3.2 

3.0 

3.2 

3.4 

3.6 

740 

4.3 

4.3 

4.2 

4.0 

3.8 

3.7 

35 

3.2 

3.0 

30 

2.9 

2.8 

2.9 

3.1 

760 

5.0 

4.7 

4.4 

4.3 

4.1 

3.8 

3.7 

3.4 

3.1 

3.0 

2.9 

2.7 

2.7 

2.8 

780 

5.3 

5.1 

4.7 

4.6 

4.4 

4.4 

4.0 

3.8 

3.4 

3.2 

2.9 

2.8 

2.7 

2.5 

800 

5.8 

5.5 

5.4 

4.8 

4.7 

4.7 

4.5 

4.2 

3.9 

3.5 

3.3 

2.9 

2.8 

2.7 

820 

6.5 

6.1 

5.8 

5.6 

5.0 

5.0 

4.9 

4.6 

4.3 

4.1 

3.6 

3.3 

3.0 

29 

840 

6.9 

6.7 

6.3 

6.1 

5.8 

5.3 

5.2 

4.9 

4.9 

4.5 

4.2 

3.9 

3.5 

3.1 

860 

7.7 

7.4 

6.9 

6.6 

6.2 

6.2 

5.5 

5.4 

5.2 

5.0 

4.8 

4.4 

4.1 

3.6 

880 

84 

7.9 

7.6 

7.1 

6.9 

6.4 

6.4 

5.8 

5.7 

5.4 

5.2 

50 

4.6 

43 

900 

9.3 

8.7 

8.3 

7.7 

7.4 

7.1 

6.7 

6.6 

6.1 

6.0 

5.6 

5.4 

5.2 

4.9 

9iO 

9.9 

9.3 

8.8 

8.4 

7.9 

7.7 

7.3 

6.9 

6.6 

6.3 

6.2 

6.1 

5.6 

54 

940 

10.6 

10.1 

9.5 

8.9 

8.7 

8.2 

7.8 

7.6 

7.2 

7.1 

6.5 

6.5 

6.3 

5.9  1 

960 

11.0 

10.7 

10.3 

9.7 

9.1 

8.7 

8.4 

8.0 

7.8 

7.4 

7.2 

6.9 

6.7 

6.5 

980 

11  4 

11.0 

10.6 

10.2 

9.8 

9.2 

8.9 

8.4 

8.1 

8.0 

7.6 

7.3 

7.2 

6.9 

1000 

11.5 

11  2 

11.0 

10.6 

10.0 

9.9 

9.5 

9.0 

8.6 

8.2 

8.1 

7.4 

7.6 

7.4 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

170 

ISO 

190 

200 

TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


II. 

200 

210 

220 

230 

240 

250 

260 

270 

280 

290 

300 

310 

j  320 

I 

0 

7.4 

7.2 

7.0 

6.6 

6.4 

6.2 

5.7 

53 

4.9 

4.7 

4.1 

3.8 

3.4 

20 

78 

7.2 

7.3 

7.2 

7.0 

6.6 

6.3 

6.0 

5.7 

5.3 

5.0 

4.4 

3.9 

40 

8.2 

8.1 

7.6 

7.5 

7.3 

7.2 

6.8 

6.6 

6.2 

5.9 

5.6 

5.2 

4.7 

60 

8.4 

8.0 

7.9 

7.8 

7.6 

7.5 

7.3 

7.1 

6.8 

6.4 

6.1 

5.8 

5.4 

80 

8.6 

8.5 

8.2 

8.0 

7.6 

7.7 

7.6 

7.4 

7.1 

7.0 

6.7 

6.3 

6.0 

100 

8.8 

8.5 

8.6 

8.4 

8.2 

7.6 

7.7 

7.8 

7.6 

7.3 

7.2 

6.9 

66 

120 

8.9 

8.7 

8.4 

8.4 

8.3 

8.3 

8.0 

7.9 

7.7 

7.6 

7.5 

7.3 

7.0 

1  140 

8.9 

8.7 

8.4 

8.3 

8.2 

8.1 

8.3 

8.0 

7.9 

7.8 

7.7 

7.5 

7.4 

160 

9.1 

8.9 

8.7 

8.4 

8.3 

8.3 

82 

8.1 

8.0 

7.9 

7.9 

7.7 

7.6  | 

180 

9.1 

8.8 

8.7 

8.5 

8.4 

8.2 

8.0 

8.0 

8.1 

7.9 

7.8 

8.0 

7.8  ! 

200 

8.9 

8.8 

8.6 

8.4 

8.4 

8.3 

8.1 

8.0 

7.9 

7.8 

7.8 

7.9 

7.9 

220 

8.7 

8.7 

8.6 

8.4 

8.2 

8.1 

8.0 

7.9 

7.8 

7.7 

7.7 

7.6 

7.7 

240 

8.5 

8.4 

8.5 

8.3 

8.1 

8.0 

7.8 

7.8 

7.8 

7.8 

7.8 

7.8 

7.6 

260 

8.2 

8.2 

8.1 

8.1 

8.1 

7.8 

7.8 

7.7 

7.6 

7.6 

7.6 

7.5 

7.4 

280 

7.8 

7.8 

8.0 

7.8 

7.9 

7.9 

7.7 

7.5 

7.5 

7.3 

7.3 

7.4 

7.3 

300 

7.3 

7.6 

7.5 

7.6 

7.7 

7.6 

7.6 

7.6 

7.4 

7.3 

7.1 

7.0 

7.1 

320 

6.6 

7.1 

7.3 

7.4 

7.4 

7.3 

7.4 

7.4 

73 

7.1 

7.0 

7.0 

6.8 

340 

6.3 

6.4 

6.7 

7.2 

7.1 

7.2 

7.2 

7.1 

7.1 

7.0 

6.9 

6.8 

6.8 

360 

6.1 

6.2 

6.4 

6.5 

6.9 

6.9 

7.0 

7.0 

6.9 

6.8 

6.7 

6.6 

6.5! 

380 

5.8 

6.1 

6.3 

6.4 

6.6 

6.7 

6.6 

6.6 

6.7 

6.8 

6.7 

6.6 

6.5 

400 

5.9 

6.0 

6.2 

6.3 

6.4 

6.5 

6.6 

6.6 

6.5 

6.6 

6.6 

6.5 

6.4 

420 

6.1 

6.3 

6.2 

6.4 

6.3 

6.4 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.4 

440 

6.3 

6.4 

64 

6.6 

6.5 

6.6 

6.5 

6.5 

6.5 

6.5 

6.3 

6.3 

6.2 

460 

6.7 

6.5 

6.5 

6.6 

6.7 

6.9 

6.7 

6.6 

6.6 

6.6 

6.5 

6.3 

6.2 

480 

7.1 

7.1 

7.0 

6.9 

69 

6.9 

7.0 

7.0 

6.8 

6.7 

6.6 

6.5 

6.3 

500 

7.4 

7.5 

7.4 

7.4 

7.3 

7.2 

7.3 

7.2 

7.1 

6.9 

6.8 

6.8 

6.6 

520 

7.9 

7.8 

7.8 

7.8 

7.8 

7.6 

7.6 

7.5 

7.5 

7.4 

7.1 

7.0 

6.9 

540 

8.3 

8.3 

8.3 

8.2 

8.2 

8.1 

8.0 

7.9 

7.9 

7.8 

7.6 

7.5 

7.2 

560 

8.2 

8.6 

8.4 

8.6 

8.7 

8.5 

8.5 

8.4 

8.2 

8.3 

8.2 

8.0 

7.6 

580 

8.2 

8.3 

8.6 

8.8 

8.8 

9.0 

8.9 

8.9 

8.7 

8.7 

8.6 

8.4 

8.4 

600 

7.7 

8.1 

8.5 

8.6 

8.9 

9.1 

9.1 

9.2 

9.2 

9.1 

9.0 

8.8 

8.7 

620 

7.4 

7.6 

8.0 

8.5 

8.7 

9.0 

9.2 

9.5 

9.5 

9.5 

9.4 

9.3 

9.2 

640 

6.7 

7.2 

7.5 

7.9 

8.3 

8.7 

9.0 

9.3 

9.5 

9.8 

9.8 

9.7 

9.7 

660 

6.0 

6.3 

7.0 

7.3 

7.7 

8.2 

8.7 

9.0 

9.4 

9.7 

9.8 

10.1 

10.0 

680 

4.9 

5.6 

6.0 

6.6 

7.1 

7.7 

8.1 

8.5 

9.0 

9.3 

9.8 

10.0 

10.2 

700 

4.2 

4.5 

5.2 

5.8 

6.4 

6.8 

7.4 

8.0 

8.5 

8.9 

9.2 

9.8 

10.1 

720 

3.6 

3.9 

4.3 

4.7 

5.3 

5.9 

6.6 

7.0 

7.8 

8.3 

8.8 

9.1 

9.7 

740 

3.1 

3.3 

3.6 

3.9 

4.4 

4.8 

5.6 

6.2 

6.9 

7.5 

8.0 

8.7 

9.2 

760 

2.8 

2.8 

3.0 

3.3 

3.6 

4.0 

4.4 

5.1 

5.8 

6.5 

7.2 

"8 

8.4 

780 

2.5 

2.6 

2.5 

2.7 

3.1 

3.3 

3.7 

4.1 

4.8 

5.4 

6.1 

6.D 

7.6 

800 

2.7 

2.5 

2.5 

2.5 

2.5 

2.7 

3.0 

3.4 

3.8 

4.4 

5.0 

5.6 

66 

820 

2.9 

2.6 

2.4 

2.3 

2.2 

2.3 

2.6 

2.8 

3.1 

3.4 

4.1 

4.7 

5.4 

840 

3.1 

2.8 

2.6 

2.4 

2.3 

2.2 

2.3 

2.4 

2.6 

2.8 

3.2 

3.8 

4.3 

860 

3.6 

3.3 

3.0 

2.7 

2.4 

2.3 

2.1 

2.2 

2.3 

2.5 

2.7 

3.0 

3.4 

880 

4.3 

3.8 

3.6 

3.2 

2.8 

2.5 

2.3 

2.1 

2.0 

2.2 

2.3 

2.5 

2.6 

900 

4.9 

4.6 

4.2 

3.6 

3.4 

2.9 

2.6 

2.3 

2.2 

2.2 

2.1 

2.2 

2.4 

920 

5.4 

5.1 

4.6 

4.5 

3.9 

3.5 

3.2 

2.9 

2.6 

2.2 

2.0 

2.1 

2.2 

940 

5.9 

5.7 

5.3 

4.9 

47 

4.3 

3.8 

3.4 

30 

2.7 

2.4 

2.1 

2.0 

960 

6.5 

6.2 

59 

5.5 

5.1 

4.9 

4.5 

4.0 

3.4 

3.1 

2.8 

2.4 

2.3 

•  980 

6.9 

6.8 

6.4 

6.1 

5.8 

5.4 

5.1 

4.8 

4.3 

3.9 

3.5 

3.0 

2.7 

1000 

7.4 

7.2 

7.0 

6.6 

6.4 

6.2 

5.7 

5.3 

4.9 

4.7 

4.1 

3.8 

34 

200 

210 

220 

2:30 

240 

250 

260 

270 

280 

290 

300 

310 

320 

TABLE  XXXI. 


35 


Perturbations  produced  by  Mara. 

Arguments  II.  and  IV. 

IV. 


II. 

320 

330 

340 

350 

360 

370 

380 

390 

400 

410 

420 

430 

414) 

' 

i 

0 

3.4 

2.8 

2.6 

2.4 

2.2 

2.3 

2.3 

2.5 

2.7 

2.9 

3.4 

4.0 

45 

20 

3.9 

3.5 

3.1 

2.7 

i  2.6 

2.4 

2.4 

2.3 

2.5 

2.7 

3.0 

3.3 

8.3 

40 

4.7 

4.2 

3.9 

3.5 

3.0 

2.8 

2.7 

\  2.6 

2.5 

2.6 

2.8 

2.9 

32 

60 

5.4 

5.0 

4.6 

4.2 

3.8 

3.4 

3.1 

2.8 

2.8 

2.7 

2.7 

2.7 

3.0 

80 

6.0 

5.7 

5.4 

4.8 

1  4.4 

4.0 

3.6 

3.4 

3.1 

29 

2.9 

2.9 

2.9 

100 

6.6 

6.3 

5.9 

5.6 

5.2 

4.8 

4.3 

4.0 

3.7 

35 

3.2 

3.0 

3.0 

120 

7.0 

6.9 

6.4 

6.1 

5.8 

5.3 

5.2 

4.6 

4.3 

4.0 

3.8 

3.6 

3.4 

140 

7.4 

7.2 

6.9 

6.6 

6.5 

6.1 

5.6 

5.4 

|  5.0 

4.6 

4.3 

4.0 

3.9 

160 

7.6 

7.5 

7.3 

7.0 

6.8 

6.6 

6.2 

5.9 

5.5 

5.3 

4.9 

4.6 

4.4 

180 

7.8 

7.7 

7.5 

7.4 

7.3 

6.9 

67 

6.5 

6.2 

5.8 

5.6 

5.3 

50 

200 

7.9 

7.8 

7.7 

7.6 

7.5 

7.3 

7.1 

6.9 

6.6 

6.4 

6.1 

5.6 

5.5 

220 

7.7 

7.7 

7.7 

7.8 

7.7 

7.5 

7.3 

7.2 

7.0 

6.7 

6.5 

6.2 

5.9 

240 

7.6 

7.6 

7.6 

7.6 

7.7 

7.6 

7.5 

7.3 

7.2 

7.1 

6.9 

6.6 

6.4 

260 

7.4 

7.3 

7.5 

7.5 

7.5 

7.6 

7.6 

7.5 

7.5 

7.3 

7.1 

7.0 

6.7 

280 

7.3 

7.4 

7.3 

7.3 

7.4 

7.4 

7.3 

7.4 

7.3 

7.5 

7.2 

7.1 

6.9 

300 

7.1 

7.1 

7.1 

7.0 

7.2 

7.3 

7.3 

7.3 

7.2 

7.2 

7.3 

7.2 

7.1 

320 

6.8 

fi.S 

6.9 

6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

7.1 

7.1 

7.0 

7.2 

340 

6.8 

6.7 

6.6 

6.6 

6.6 

6.8 

6.9 

6.9 

7.0 

7.0 

6.9 

6.9 

6.9 

360 

6.5 

6.5 

6.4 

6.3 

6.4 

6.5 

6.6 

6.7 

6.8 

6.8 

6.8 

6.8 

6.9 

380 

6.5 

6.3 

6.3 

6.2 

6.2 

6.2 

6.3 

6.3 

6.4 

6.5 

6.6 

6.7 

6.7 

400 

6.4 

6.2 

6.2 

6.0 

6.1 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

6.3 

6.4 

420 

6.4 

6.2 

6.1 

6.0 

5.9 

5.8 

5.9 

5.9 

5.9 

5.9 

5.9 

6.0 

6.0 

440 

6.2 

6.1 

6.0 

5.8 

5.8 

5.7 

5.6 

5.6 

5.6 

5.7 

5.7 

5.8 

5.9 

460 

6.2 

6.0 

5.9 

5.8 

5.7 

5.5 

5.5 

5.4 

5.5 

5.4 

5.5 

5.3 

5.4 

480 

63 

6.2 

6.0 

5.7 

5.6 

5.5 

5.4 

5.3 

5.2 

5.2 

5.2 

5.3 

5.3 

500 

6.6 

6.4 

6.2 

6.0 

5.7 

5.4 

5.3 

5.2 

5.1 

5.1 

5.1 

5.0 

5.0 

520 

6.9 

6.7 

6.4 

6.1 

6.1 

5.7 

5.5 

5.1 

5.1 

5.0 

4.9 

5.0 

4.9 

540 

7.2 

7.1 

6.7 

6.5 

6.2 

6.1 

5.8 

5.5 

5.2 

5.0 

4.9 

4.8 

4.8 

560 

7.6 

7.4 

7.3 

7.0 

6.6 

6.3 

6.0 

5.8 

5.4 

5.3 

5.0 

4.7 

4.7 

580 

8-4 

8.0 

7.8 

7.5 

7.0 

6.8 

6.3 

6.2 

5.9 

5.5 

5.3 

5.0 

4.9 

600 

8.7 

8.6 

8.3 

8.0 

7.8 

7.4 

7.0 

6.6 

6.3 

6.0 

5.6 

5.3 

5.1 

620 

9.2 

9.1 

8.9 

8.6 

8.4 

8.1 

7.6 

7.2 

6.8 

6.5 

6.1 

5.7 

5.3 

640 

9.7 

9.G 

9.4 

9.3 

9.0 

8.7 

8.2 

7.8 

7.4 

7.0 

6.6 

6.3 

5.8 

660 

10.0 

10.0 

9.9 

9.8 

9.6 

9.3 

8.9 

8.5 

8.2 

7.7 

7.2 

6.8 

6.4 

680 

10.2 

10.4 

10.3 

10.2 

10.1 

9.9 

9.6 

9.3 

9.0 

8.5 

8.1 

7.5 

7.1 

700 

10.1 

10.3 

10.5 

10.6 

10.4 

10.3 

10.1 

9.8 

9.6 

9.3 

8.9 

8.3 

7.8 

720 

9.7 

10.1 

10.3 

10.6 

10.7 

10.6 

10.5 

10.5 

10.2 

10.0 

9.6 

9.2 

8.6 

740 

9.2 

9.6 

10.0 

10.3 

10.6 

10.7 

10.8 

10.9 

10.6 

:o.5 

10.2 

9.9 

9.4 

760 

8.4 

9.0 

9.5 

9.8 

10.2 

10.6 

10.9 

11.0 

11.0 

11.0 

10.7 

10.5 

10.3 

780 

7.6 

8.2 

8.9 

9.4 

9.9 

10.3 

10.6 

10.9 

11.1 

11.2 

11.0 

10.81 

10.7 

800 

6.6 

7.3 

7.9 

8.5 

9.2 

9.8 

10.1 

10.6 

10.8 

11.1 

11.3 

11.1 

11.0 

820 

5.4 

6.0 

7.0 

7.6 

8.2 

8.9 

9.6 

10.0 

10.5 

10.8 

11.0 

11.3 

11.3 

840 

4.3 

5.0 

5.6 

6.5 

7.2 

7.9 

9.8 

9.2 

9.9 

10.3 

107 

10.9 

11.2 

860 

3.4 

4.0 

4.6 

5.3 

6.1 

6.9 

7.5 

8.4 

9.1 

9.6 

10.1 

10.7 

10.9 

880 

2.6 

3.1 

3.7 

4.3 

5.0 

5.7 

6.6 

7.1 

8.1 

8.7 

9.4 

9.8 

104 

900 

2.4 

2.7 

3.0 

3.4 

4.0 

4.6 

5.4 

6.1 

6.9 

7.6 

8.4 

9.1 

9.7 

920 

2.2 

2.3 

2.3 

2.8 

3.3 

3.7 

4.3 

5.0 

5.8 

6.5 

7.2 

8.0 

8.7 

040 

2.0 

2.1 

2.3 

2.3 

2.7 

2.9 

3.4 

4.1 

4.7 

5.5 

6.1 

7.0 

7.7 

960 

2.3 

2.2 

2.2 

2.3 

2.3 

2.5 

2.8 

3.2 

3.9 

4.5 

5.1 

5.7 

65 

980 

•2.7 

2.4 

2.2 

2.3 

2.3 

2.4; 

2.5 

2.8 

3.0 

36 

4,1 

4.7 

55 

1000 

3.4 

2.8 

2.6 

2.4 

2.2 

2.3 

2.3 

2.5 

27 

2.9 

3.4 

4.0 

45 

320 

330 

340 

3cO 

360  ; 

370 

380 

390 

400 

410 

420 

430 

440 

36 


TABLE  XXXI. 

Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


n. 

440 

450 

460 

470 

480 

490 

500 

510 

520 

530 

540 

550 

560 

0 

4.5 

5.2 

5.9 

6.6 

7.3 

8.0 

8.5 

9.0 

9.5 

10.0 

10.4 

10.7 

10.9 

20 

3.8 

4.3 

4.9 

5.6 

6.2 

6.9 

7:6 

8.2 

8.8 

9.3 

9.7 

10.0 

11.4 

40 

3.2 

3.7 

4.2 

4.8 

5.4 

5.9 

6.6 

7.3 

7.9 

8.4 

8.9 

9.4 

9.8 

60 

3.0 

3.2 

3.6 

4.0 

4.5 

5.1 

5.7 

6.3 

6.9 

7.5 

8.0 

8.6 

9.1 

80 

2.9 

3.1 

3.3 

3.5 

3.9 

4.4 

4.9 

5.4 

5.9 

6.5 

7.1 

7.7 

8.2 

100 

'  3.0 

3.1 

3.2 

3.5 

3.6 

3.8 

4.2 

4.8 

5.3 

5.9 

6.4 

6.9 

7.4 

120 

3.4 

3.3 

3.3 

3.4 

3.5 

3.6  | 

3.9 

4.2 

4.7 

5.1 

5.6 

6.0 

6.6 

140 

3.9 

3.8 

3.6 

3.6 

3.6 

3.7 

4.0 

4.0 

4.2 

4.6 

5.0 

5.4 

5.9 

160 

4.4 

4.2 

3.9 

4.1 

3.8 

3.7 

4.0 

4.1 

4.2 

4.5 

4.6 

4.9 

5.3 

180 

5.0 

4.8 

4.4 

4.2 

4.2 

4.2 

4.0 

4.1 

4.3 

4.4 

4.4 

4.7 

5.0 

200 

5.5 

5.2 

5.1 

4.8 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.7 

4.6 

4.8 

220 

5.9 

5.7 

5.5 

5.3 

5.1 

4.9 

4.9 

4.8 

4.7 

4.8 

4.8 

4.9 

5.0 

240 

6.4 

6.2 

5.9 

5.8 

5.6 

5.4 

5.3 

5.2 

5.1 

5.1 

5.1 

5.2 

5.2 

260 

6.7 

6.6 

6.4 

6.1 

6.0 

5.9 

5.8 

5.7 

5.6 

5.5 

5.4 

5.4 

5.4 

280 

6.9 

6.8 

6.7 

6.5 

6.3 

6.2 

6.1 

6.0 

5.9 

5.9 

5.9 

5.8 

5.8 

300 

7.1 

7.0 

6.8 

6.8 

6.6 

6.5 

6.4 

6.3 

6.2 

6.2 

6.* 

6.2 

6.2 

320 

7.2 

7.1 

6.9 

6.8 

6.8 

6.7 

6.6 

6.5 

6.5 

6.5 

6.5 

6.6 

6.61 

340 

6.9 

6.9 

7.0 

6.9 

6.9 

6.8 

6.7 

6.8 

6.7 

6.6 

6.7 

6.8 

6.9 

360 

6.9 

6.8 

6.8 

6.8 

6.8 

6.7 

6.7 

6.6 

6.6 

6.8 

6.8 

6.8 

6.9 

380 

6.7 

6.5 

6.5 

6.6 

6.7 

6.6 

6.6 

6.7 

6.7 

6.7 

6.8 

6.9 

6.9 

400 

6.4 

6.4 

6.3 

6.3 

6.4 

6.5 

6.5 

6.5 

6.6 

6.7 

6.7 

6.S 

68 

420 

6.0 

62 

6.3 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.5 

6.6 

6.7 

440 

5.9 

5.9 

6.0 

6.0 

6.0 

6.0 

6.0 

6.1 

6.0 

6.1 

6.2 

6.2 

6.4 

460 

5.4 

5.5 

5.7 

5.8 

5.8 

6.8 

5.8 

5.8 

5.8 

5.8 

5.9 

6.0 

6.1 

480 

53 

5.3 

5.5 

5.5 

5.5 

5.6 

5.5 

5.6 

5.4 

5.6 

5.7 

5.5 

5.8 

500 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.3 

5.4 

5.4  1 

520 

4.9 

4.9 

4.9 

4.8 

5.0 

5.1 

5.1 

5.1 

5.1 

5.1 

5.0 

5.0 

5.1 

540 

4.8 

4.8 

4,7 

4.8 

4.8 

4.9 

4.9 

5.0 

4.9 

4.8 

4.8 

4.9 

4.8 

560 

4.7 

4.6 

4.6 

4.7 

4.7 

4.6 

4.7 

4.7 

4.7 

4.7 

4.6 

4.6 

4.6 

580 

4.9 

4.6 

4.5 

4.5 

4.6 

4.5 

4.4 

4.4 

4.5 

4.5 

4.5 

4.4 

4.4 

600 

5.1 

4.9 

4.6 

4.5 

4.4 

4.4 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.3 

620 

5.3 

5.1 

4.9 

4.7 

4.6 

4.4 

4.3 

4.1 

4.2 

4.2 

4.2 

,  4.2 

4.1 

640 

5.8 

5.4 

5.2 

5.0 

4.7 

4.6 

4.4 

i  4.1 

4.1 

4.1 

4.2 

4^2 

4.0 

660 

6.4 

6.0 

5.7 

5.4 

5.0 

4.8 

4.7 

4.5 

4.3 

4.2 

4.2 

4.1 

4.0 

680 

7.1 

6.6 

6.2 

5.7 

5.4 

5.1 

4.9 

4.7 

4.5 

4.4 

4.3 

4.0 

3.9 

700 

7.8 

7.2 

6.8 

6.4 

6.0 

5.6 

5.3 

5.0 

4.7 

4.6 

4.6 

4.3 

4.1 

720 

8.6 

8.0 

7.6 

7.1 

6.6 

6.2 

5.7 

5.5 

5.2 

4.9 

4.6 

4.6 

4.3 

740 

9.4 

9.0 

8.4 

8.0 

7.4 

6.9 

6.3 

6.0 

5.6 

5.3 

5.0 

4.7 

4.5 

760 

10.3 

9.7 

9.3 

8.6 

8.1 

7.6 

7.2 

6.5 

6.2 

5.8 

5.5 

5.2 

4.9 

780 

10.7 

10.5 

i  9.9 

9.6 

9.0 

8.5 

7.8 

7.4 

7.0 

6.4 

6.1 

5.7 

5.5 

800 

11.0 

11.0 

10.6 

10.2 

9.9 

9.3 

8.8 

8.1 

7.7 

7.3 

6.7 

6.3 

5.8 

820 

11.3 

11.1 

10.9 

10.6 

10.3 

10.0 

9.6 

9.1 

8.5 

7.9 

7.4 

7.0 

6.6 

840 

11.2 

111.3 

11.2 

11.1 

11.0 

10.7 

10.2 

9.9 

9.4 

8.8 

8.2 

7.7 

7.3 

860 

10.9 

11.1 

11.4 

11.3 

11.3 

11.2 

10.7 

10.4 

9.9 

9.6 

9.2 

8.5 

7.9 

880 

10.4 

10.8 

11.0 

11.3 

11.2 

11.2 

11.2 

10.9 

10.5 

10.3 

9.8 

9.3 

8.7 

900 

9.7 

10.1 

10.6 

11.0 

11.2 

11.2 

11.2 

11.0 

10.9 

10.7 

10.2 

10.0 

9.4 

920 

8  7 

9.3 

9.9 

10.3 

10.8 

11.0 

11.1 

11.2 

11.2 

11.0 

10.7 

10.4 

10.1 

940 

77 

8.2 

8.8 

9.5 

10.1 

10.4 

10.9 

11.0 

11.2 

11.2 

11.0 

10.7 

10.5 

960 

I  <?.5 

7.3 

8.1 

8.6 

9.3 

9.8 

10.2 

:  10.6 

10.8 

11.1 

11.2 

10.9 

10.8 

980 

!  5.5 

6.2 

7.0 

7.7 

8.3 

8.9 

9.5 

,  10.0 

10.4 

10.6 

10.8 

11.0 

10.9 

1000 

4.5 

5.2 

5.9 

6.6 

7.3 

8.0 

8.5 

;  9.0 

9.5 

100 

10.4 

10.7 

10.9 

440 

450 

460 

470 

I  480 

.490 

i  500 

;  510 

520 

530 

540 

550 

560 

TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II  and  IV. 

IV. 


11. 

560  570 

580 

590  600 

|  610  620 

630 

640 

650 

660  f  670 

680 

0 

10  9J  10.8 

10.6 

104  10.3 

10.0 

9.7 

9.2 

8.9 

8.5 

8.1 

7. 

7.7 

20 

11.4  10.6 

10.7 

10.6  104  10.2  9.9 

9.7 

9.3 

9.0  8.8 

8. 

8.1 

40 

9.8  i  10.1 

10.4 

10.4  10.5  10.3  10.2 

9.9 

9.6 

9.4  9.1 

8. 

8.5 

60 

9.1  1  9.4 

9.8 

10.2  10.2  1  10.3  10.2 

10.1 

9.9 

9.6!  93 

9. 

8.8 

80 

8.2:  8.7 

9.0 

93   9.6 

9.8 

100 

9.9 

9.8 

9.7 

9.5 

9. 

91 

100 

7.4   7.9 

8.4 

8.7;  9.0 

9.4 

9.6 

9.7 

9.8 

9.7 

9.7 

9. 

9.2 

120 

6.6   69 

7.6 

8.1  83 

8.6 

9.0 

92 

9.4 

9.5l  9.5 

9.4 

9.3 

140 

5.9   6.3  6.8   7.2!  7.7 

8.0  8.3 

8.7 

8.9 

9.1  9.2  9.3 

9.3 

160 

5.:*   5.8 

6.0 

65   6.9 

7.4  7.7 

8.0 

8.4 

8.5!  8.8  :  8.9 

9.0' 

180 

5.0   5.2 

5.6 

60j  6.3  1  6.7  7.1 

7.2 

7.7 

8.1 

8.1  i  8.4 

8.6 

200 

4.8   5.0 

5.3 

5.4 

5.8 

6.1  |  6.5 

6.7 

7.1 

7.3 

7.7   7.8 

8.0 

220 

5.0!  5.0 

5.1 

5.3 

5.5 

5.7  j  6.0 

6.3 

6.6 

6.8 

7.0 

7.3 

7.5 

240 

5.2 

5.2 

5.3 

5.3 

54 

5.5 

5.7 

5.9 

6.1 

6.4!  6.6 

6.8 

7.1 

260 

5.4   5.5 

5.5 

5.5 

5.5 

5.5 

5.5 

5.7 

5.8 

6.0 

6.3 

6.4 

6.5 

280 

5.8 

5.8 

5.8 

5.9 

5.8 

5.8 

5.8 

5.9 

5.9 

5.9 

6.0 

6.1 

6.2 

300 

6.2 

6.1 

6.2 

6.1 

6.1 

6.1 

6.2 

6.1 

6.0 

5.9  5.9 

6.0 

6.1  j 

320 

6.6 

6.5 

6.6 

6.6 

6.5 

6.5 

6.6 

6.5 

6.5 

6.3 

6.1 

6.0 

6.0 

340 

6.9 

6.9 

6.9 

7.0 

7.0 

6.9 

6.8 

6.9 

6.9 

6.8 

6.6 

6.5 

6.3 

360 

6.9 

7.0 

7.2 

7.3 

7.3 

7.3 

7.4 

7.3 

7.3 

7.1 

7.1 

7.0 

6.7 

380 

6.9 

7.0 

7.2 

7.4 

7.5 

7.6 

7.7 

7.7 

7.7 

7.6 

7.5 

7.4 

7.2 

400 

6.8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.0 

8.0 

8.1 

8.1 

8.1 

7.9 

7.8 

420 

6.7 

6.9 

7.0 

7.2 

7.6 

7.8 

8.0 

8.2 

8.3 

8.4 

8.4 

8.5 

8.4 

440 

6.4 

6.6 

6.9 

7.0 

7.3 

7.5 

7.9 

8.2 

8.4 

8.6 

8.8 

8.8 

8.9 

460 

6.1 

6.2 

6.5 

6.9 

7.1 

7.2 

7.6 

8.0 

8.4 

8.7 

9.0 

9.1 

9.2 

480 

5.8 

5.9 

6.0 

6.2 

6.7 

7.1 

7.2 

7.6 

7.9 

8.5 

8.9 

9.2 

9.3 

500 

5.4 

5.5 

5.6 

5.9 

6.1 

6.4 

6.9 

.  7.2 

7.7 

7.9 

8.4 

9.0 

9.4 

520 

5.1 

5.2 

5.2 

5.3 

5.6 

5.9 

6.3 

6.7  7.0 

7.6 

8.0 

8.4 

9.0 

540 

4.8  4.8 

4.8 

5.0 

5.1 

5.4 

5.6 

6.0  6.4 

6.7 

7.5 

8.1 

8.5 

560 

4.6  |  4.5 

4.5 

4.5 

4.7 

4.8 

5.0 

5.3 

5.8 

6.2 

6.6 

7.1 

7.8 

5SO 

4.4 

4.3 

4.3 

4.3 

4.3 

4.3 

4.5 

4.7 

5.2 

5.5 

5.9 

6.4 

6.9 

600 

4.3 

4.3 

4.2 

4.1 

4.0 

4.0 

4.1 

4.2 

4.5 

4.8 

5.1 

5.7 

6.2 

620 

4.1 

4.0 

4.0 

3.9 

3.9 

3.8 

3.8 

3.8 

3.8 

4.0 

4.4 

4.9 

5.4 

640 

4.0 

3.9 

4.0 

3.8 

3.8 

3.8 

3.7 

3.5 

3.5 

3.6 

3.8 

4.0 

4.5 

660 

4.0 

4.0 

3.9 

3.8 

3.7 

3.5 

3.5 

3.4 

3.3 

3.3 

3.4 

3.5 

3.7 

6SO 

3.9 

4.0 

39 

3.8 

36 

3.5 

3.4 

3.3 

3.2 

3.1 

3.1 

3.1 

3.1 

700 

4.1 

3.9 

3.9 

•  3.9 

3.7 

3.5 

3.4 

3.3 

3.2 

3.0 

3.0 

3.0 

2.9 

720 

4.3 

4.1 

4.0 

3.9 

3.8 

3.8 

3.5 

3.4 

3.1 

2.9 

2.9 

2.7 

2.7 

740 

4.5 

4.2 

4.2 

4.2 

4.0 

3.7 

3.6 

3.4 

3.3 

3.0 

2.8 

2.6 

2.5 

760 

4.9 

4.7 

4.5 

4.3 

4.2 

4.1 

3.8 

3.6 

3.3 

3.1 

2.9 

2.8 

2.5 

780 

5.5 

5.1 

4.9 

4.5 

4.4 

4.3 

4.1 

3.9 

3.8 

3.4 

3.2 

3.0 

2.7 

800 

5.8 

5.6 

5.2 

5.0 

4.6 

4.5 

4.4 

4.3 

4.1 

3.8 

3.5 

3.1 

2.8 

820 

6.6 

6.1 

5.8 

5.5 

5.3 

5.0 

4.8 

4.6 

4.4 

4.2 

4.0 

3.6 

3.3 

840 

7.3 

6.8 

6.5 

6.1 

5.7 

5.5 

5.2 

5.0 

4.7 

4.6 

4.3 

4.1 

3.8 

860 

7.9 

7.5 

7.0 

6.7 

6.4 

5.9 

5.8 

5.4 

5.1 

5.0 

4.8 

4.6 

4.4 

880  i  8.7 

8.2 

7.8 

7.3 

6.9 

6.6 

6.3 

6.0 

5.7 

5.4 

5.2 

5.0 

4.7 

900  |  9.4 

9.0 

8.5 

8.0 

7.6 

7.2 

6.8 

6.6 

6.3 

5.9 

5.6 

5.4 

5.2 

920  10.1 

9.8 

9.2 

8.7 

8.3 

7.8 

7.4 

7.0 

6.7 

6.4 

6.0 

5.8 

5.7 

940 

10.5 

10.2 

9.8 

9.4 

8.8 

8.5 

8.0 

7.6 

7.3 

6.9 

6.6, 

62 

6  1 

960 

10.8  10.5 

10.2 

10.0 

9.5 

9.1 

8.6 

8.2 

7.8 

7.5!  7.1 

6.8 

66 

980 

10.9  10.7 

10.3 

10.2 

9.9 

9.6  9.2 

9.0 

8.5 

8.0  7.7 

7.4 

72 

1000 

10.9  !  10.8 

10.6 

10.4 

10.3 

10.0  ;  9.7 

9.2 

8.9 

8.5 

8.1 

7.9 

7.7 

i 

560  570 

580 

590 

600 

610  '  620 

630  640 

650 

6PO 

670 

680 

31 


38 


TABLE  XXXI. 


Perturbations  produced  by  Mars. 

Arguments  II.  and  IV. 

IV. 


1L 

680 

090 

700 

710 

720 

730 

740 

750 

760 

770 

780 

790 

800 

0 

7.7 

7.4 

6.9 

6.8 

6.7 

6.4 

6.1 

5.8 

5.5 

5.2 

4.8 

4.4 

3.7 

20 

8.1 

7.8 

7.4 

7.0 

7.1 

6.9 

6.7 

6.4 

6.1 

5.8 

5.5 

5.1 

4.7 

40 

8.5 

8.3 

7.8 

7.5 

7.2 

7.1 

7.0 

6.9 

6.6 

6.4 

6.1 

5.8 

5.3 

60 

8.8 

8.6 

8.3 

8.1 

7.8 

7.6 

7.5 

7.4 

7.1 

6.9 

6.7 

6.3 

6.0 

80 

9.1 

8.9 

8.7 

8.4 

8.1 

8.0 

7.8 

7.6 

7.4 

7.3 

7.1 

6.9 

6.5 

100 

9.2 

8.9 

8.8 

8.7 

8.6 

8.3 

8.0 

7.7 

7.6 

7.6 

7.6 

7.3 

7.0 

120 

9.3 

9.2 

9.0 

8.7 

8.6 

8.4 

8.2 

8.1 

7.9 

7.8 

7.7 

7.6 

7.5 

140 

93 

9.2 

9.0 

9.0 

8.7 

8.5 

8.4 

8.3 

8.0 

7.8 

7.7 

7.7 

7.7 

160 

9.0 

9.0 

8.9 

8.8 

8.7 

8.6 

8.5 

8.4 

8.2 

8.0 

7.9 

7.8 

7-8 

180 

8.6 

8.6 

8.7 

8.7 

8.7 

8.6 

8.5 

8.3 

8.3 

8.0 

8.2 

7.8 

7.9 

200 

8.0 

8.2 

8.3 

8.3 

8.5 

8.4 

8.4 

8.4 

8.2 

8.1 

8.1 

8.1 

7.9 

220 

7.5 

7.7 

7.9 

8.1 

8.2 

8.2 

8.1 

8.2 

8.2 

8.0 

8.1 

8.0 

8.0 

240 

7.1 

7.2 

7.4 

7.5 

7.6 

7.7 

7.8 

7.8 

7.9 

8.0 

8.0 

7.8 

7.8 

260 

6.5 

6.7 

6.9 

7.1 

7.2 

7.3 

7.4 

7.5 

7.6 

7.6 

7.7 

7.7 

7.8 

280 

6.2 

6.3 

6.5 

6.7 

67 

6.9 

7.1 

7.2 

7.3 

7.3 

7.3 

7.3 

7.4 

300 

6.1 

6.0 

6.2 

6.4 

6.4 

6.5 

6.6 

6.7 

6.9 

6.9 

6.9 

7.1 

7.1 

320 

6.0 

6.0 

6.0 

6.0 

6.2 

6.1 

6.2 

6.3 

6.5 

6.5 

6.6 

6.6 

6.8 

340 

6.3 

6.2 

6.0 

6.0 

6.0 

6.0 

6.1 

6.1 

6.2 

6.2 

6.3 

6.3 

6.4 

360 

67 

6.6 

6.4 

6.1 

6.0 

5.9 

6.0 

5.9 

5.9 

5.9 

6.0 

6.1 

6.2 

380 

7.2 

7.1 

6.8 

6.6 

6.4 

6.2 

6.1 

5.9 

5.8 

5.7 

5.6 

5.8 

5.9 

400 

7.8 

7.7 

7.4 

7.1 

6.8 

6.6 

6.4 

6.1 

6.0 

5.8 

5.6 

5.5 

5.6 

420 

8.4 

8.2 

8.0 

r.8 

7.5 

7.2 

6.8 

6.5 

6.2 

6.0 

5.7 

5.5 

5.4 

440 

8.9 

8.8 

8.7 

8.4 

8.2 

7.8 

7.5 

7.1 

6.6 

6.2 

6.0 

5.7 

5.6 

460 

9.2 

9.2 

9.2 

9.0 

8.8 

8.5 

8.2 

7.9 

7.5 

6.9 

6.5 

6.3 

6.0 

480 

9.3 

9.5 

9.6 

9.6 

9.4 

9.2 

9.1 

8.6 

8.3 

7.8 

7.2 

6.9 

6.5 

500 

9.4 

9.6 

9.8 

100 

9.9 

9.8 

9.6 

9.4 

9.1 

8.7 

8.2 

7.6 

7.2 

520 

9.0 

9.5 

9.8 

10.1 

10.2 

10.3 

10.3 

10.0 

9.8 

9.5 

9.1 

8.5 

8.0 

540 

8.5 

9.1 

9.5 

10.0 

10.3 

10.5 

10.6 

10.6 

10.4 

10.1 

9.8 

9.5 

9.0 

560 

7.8 

8.5 

9.0 

9.5 

9.9 

10.4 

10.8 

10.8 

10.9 

10.8 

10.6 

10.2 

9.9 

580 

6.9 

7.6 

8.3 

9.0 

9.7 

10.0 

10.4 

10.7 

11.1 

11.2 

11.0 

11.0 

10.6 

600 

6.2 

6.8 

7.4 

8.0 

8.9 

9.6 

10.1 

10.4 

10.9 

11.3 

11.4 

11.3 

11.2 

620 

5.4 

5.9 

6.5 

7.1 

7.8 

8.6 

9.4 

10.3 

10.6 

11.0 

11.5 

11.7 

11.7 

640 

4.5 

5.0 

5.5 

6.2 

68 

7.6 

8.4 

9.2 

10.0 

10.7 

11.1 

11.6 

11.8 

660 

3.7 

4.1 

4.7 

5.2 

5.9 

6.5 

7.3 

8.3 

9.1 

9.8 

10.5 

11.2 

11.5 

680 

3.1 

3.4 

3.8 

4.3 

4.8 

5.5 

6.2 

7.0 

7.8 

8.7 

9.6 

1C.2 

11.0 

700 

2.9 

2.8 

30 

3.4 

3.9 

4.5 

5.2 

6.0 

6.7 

7.5 

8.C 

9.4 

10.1 

720 

2.7 

2.6 

2.5 

2.7 

3.1 

3.5 

4.0 

4.8 

5.6 

6.4 

7.3 

8.2 

9.1 

740 

2.5 

2.4 

2.4 

2.4 

25 

2.7 

3.1 

3.6 

4.5 

5.2 

6.1 

6.9 

7.8 

760 

2.5 

2.3 

2.2 

2.1 

2.1 

2.3 

2.4 

2.8 

3.2 

4.1 

4.7 

5.7 

6.6 

780 

2.7 

2.5 

2.3 

2.1 

2.0 

1.9 

2.1 

2.2 

2.5 

2.9 

3.6 

4.4 

5.2 

800 

2.8 

2.7 

2.4 

2.2 

2.0 

1.8 

1.8 

1.8 

2.0 

2.3 

2.5 

3.2 

4.0 

S20 

3.3 

3.0 

2.7 

2.3 

2.1 

1.9 

1.8 

1.5 

1.7 

1.7 

2.0 

2.2 

2.9 

840 

3.8 

3.5 

3.0 

2.6 

2.3 

2.1 

1.9 

1.6 

1.5 

1.5 

1.6 

1.7 

2.2 

860 

4.4 

4.0 

3.5 

32 

2.8 

2.3 

1.9 

1.7 

1.4 

1.3 

1.2 

1.4 

1.6 

880 

4.7 

4.4 

.4.1 

3.7 

3.3 

3.0 

2.5 

2.1 

1.7 

1.4 

1.3 

1.2 

1.2 

900 

5.2 

5.0 

4.6 

4.3 

4.0 

3.6 

3.2 

2.7 

2.2 

l.G 

1.3 

1.2 

1.1 

920 

5.7 

5.3 

5.1 

5.0 

46 

4.2 

3.8 

3.4 

2.9 

23 

1.9 

1.3 

1.1 

940 

6.1 

5.9 

5.6 

5.4 

5.2 

4.8 

4.5 

3.9 

3.5 

3.1 

2.6 

2.1 

1.5 

i  960 

6.6 

6.4 

6.2 

5.8 

5.6 

5.4 

5.1 

4.7 

4.3 

3.7 

3.2 

2.8 

2.3 

j  .980 

1  7.2 

6.9 

i  6.6 

6.4 

6.2 

5.9 

5.6 

5.3 

5.0 

4.6 

4.0 

3.5 

3.0 

'1COO 

7.7 

7.4 

6.9 

6.8 

6.7 

6.4 

6.1 

5.8 

5.5 

52 

4.8 

4.4 

3.7 

680 

690 

700 

710 

720 

730 

740 

750 

760 

770 

780 

790 

800 

TABLE  XXXI. 

Perturbations  produced  by  Mars 

Arguments  II.  and  IV. 

IV. 


39 


II. 

800 

810 

820  830  840  ,  850  860 

870  880  ;  890 

900 

910 

«£0 

0 

3.7 

3.2 

2.6 

2.1   1.7 

1.3  0.9 

0.7 

0.7 

1.0 

1.2 

1.6 

22 

20 

4.7 

4.2 

3.6 

3.1   2.4 

1.9  1.5 

1.2 

0.8  0.6 

0.9   1.2 

5 

40 

5.3 

4.9  4.5 

38  33 

2.7J  2.0 

1.7 

1.4  1.0 

0.8  0.9 

.0 

60   6.0  5.7   5.2   4.7  4.1   3.6  3.1 

2.6 

2.0   1.5 

1.2  0.9 

.0 

80  !  6.n 

6.3!  6.0 

5.5   5.0 

4.6 

4.0 

3.4 

2.7  2.2 

1.8 

1.5 

3 

100 

7.0 

6.7 

6.5 

6.3  j  5.9 

5.3 

4.9 

4.4 

3.7 

3.1 

2.5,  2.1 

.7 

120 

7.5 

7.3 

7.0 

6.8  6.5 

6.2 

5.7 

5.1 

4.7 

4.1 

3.5 

!  2.9 

24 

140 

7.7 

7.7 

7.5 

7.3 

7.0 

6.7 

6.4 

6.0 

5.6 

5.1 

4.5 

i  3.8 

33 

160 

7.8 

7.9 

7.7 

7.6 

7.4 

7.2 

7.0 

6.8 

6.3 

5.8 

5.4 

4.8 

42 

180 

7.9 

7.8 

7.9 

7.9 

7.7 

7.6 

7.5 

7.1 

7.0 

6.6 

6.1 

5.7 

5.2 

200 

7.9 

7.9 

7.8 

7.9 

7.8 

7.7 

7.6 

7.5 

7.5 

7.1 

6.8 

6.3 

6.1 

220 

8.0 

7.9 

7.8 

7.8 

7.8 

7.8 

7.8 

7.8 

7.6 

7.5 

7.4 

7.1 

6.7 

240 

7.8 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 

7.8 

7.7 

7.6 

7.6 

7.5 

7.2 

260 

7.8 

7.7 

7.7 

7.6 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 

7.8 

7.6 

280 

7.4 

74 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.5 

7.6 

7.6 

7.8 

7.7 

300  i  7.1 

72 

7.3   7.3 

7.3 

7.3 

7.3 

7.4 

7.5 

7.4 

7.5 

7.5 

7.7 

3201  '68J  6.9 

6.8 

7.0 

7.1 

7.1 

7.1 

7.1 

73 

7.3 

7.3 

7.4 

7.4 

340 

64 

6.5 

6.6 

6.6 

6.7 

6.7 

6.8 

6.9 

7.0 

7.1 

7.2 

7.2 

7.2 

360 

6.2  6.2 

6.2 

6.3 

6.4 

6.4 

6.5 

6.6 

6.7 

6.7 

6.9 

6.9 

7.1 

380 

o  91  5.8 

5.8 

5.9 

6.0 

6.1 

6.2 

6.3 

6.4 

6.4 

6.4 

6.6 

6.8 

400 

56 

5.6 

5.6 

5.7 

5.7 

5.7 

5.8 

5.9 

5.9 

6.0 

6.1 

6.2  64 

420 

54 

5.4 

5.5 

5.5 

5.5 

5.5 

5.5 

5.5 

5.6 

5.6 

5.6 

5.7 

5.8 

440 

56!  5.3 

5.3 

5.3 

5.3 

5.2 

5.2 

5.2 

5.2 

5.1 

5.0 

5.3 

5.5 

460 

60  5.6 

5.4 

5.3 

5.2 

5.2 

5.1 

5.0 

5.1 

5.2 

5.2 

5.2 

53 

480 

6.5 

6.0 

5.7 

5.4 

5.2 

5.2 

5.1 

4.9 

4.9 

49 

4.9 

5.0 

50 

500 

7.2 

6.8 

6.3 

5.9 

5.6 

5.3 

5.0 

4.8 

4.9 

4.8 

4.8 

4.8 

49 

520 

8.0 

7.4 

7.0 

6.5 

6.1 

5.5 

5.4 

5.1 

4.9 

4.7 

4.7 

4.7 

4.8 

540 

9.0  8.4 

7.8 

7.3 

6.7 

6.3 

5.8 

5.4 

5.2 

4.9 

4.7 

4.7 

4.7 

560 

9.9  9.5 

8.8 

8.2 

7.7 

7.1 

6.5 

6.0 

5.7 

5.3 

5.0 

4.8 

4.6 

580 

10.6  10.2 

9.8 

9.3 

8.8 

8.1 

7.2 

6.8 

6.4 

6.0 

5.6 

5.1 

4.9 

600 

11.21  11.0 

10.7 

10.3 

9.6 

9.1 

8.5 

7.7 

7.1 

6.4 

6.1 

5.6 

5.3 

620 

11.7 

11.5 

11.4 

11.0 

10.6 

9.9 

9.5 

8.9 

8.1 

7.4 

6.8 

6.3 

5.9 

640 

11.8  11.9 

11.8 

11.7 

11.3 

11.0 

10.4 

9.8 

9.3  !  8.5 

7.8 

7.1 

6.6 

660 

11.5  11.8 

12.0 

12.1 

11.9 

11.6 

11.2 

0.8 

10.2 

9.6 

8.9 

8.2 

7.5 

680 

11.01  11.6 

12.1 

12.2 

12.1 

12.2 

12.1 

1.5 

11.1 

10.6 

10.1 

9.2 

8.5 

700 

10.1 

10.9 

11.6 

12.1 

12.4 

12.3 

12.3 

2.3 

11.9 

11.4 

10.8 

10.4,  9.7 

720 

9.1 

10.0 

10.6 

11.4 

11.9 

12.4 

12.6 

2.5 

12.4 

12.0 

11.6 

11.2  10.8 

740 

7.8 

8.8 

9.7 

10.5 

11.3 

11.8 

12.3 

2.8 

12.6 

12.6 

12.3 

11.9  11.5 

760 

6.6 

7.6 

8.5 

9.4 

10.3 

11.0 

11.7 

2.1 

12.6 

12.8 

12.7 

12.5;  12.1 

780 

5.2 

6.3 

7.1 

8.1 

9.2 

10.1 

10.7 

1.6 

12.0 

12.4 

12.8 

12.9  j  12.8 

800 

4.0 

4.8 

5.7 

6.7 

7.7 

8.7 

9.7 

0.5 

11.3 

11.9 

12.3 

12.5  12.9 

820 

2.9 

3.6 

4.4 

5.4 

6.4 

7.3 

8.4 

9.5  103 

11.0 

11.7 

12.1  |  12  5 

840 

2.2 

2.7 

3.3 

4.0 

4.9 

6.0  i  7.0 

8.0 

9.1 

10.0 

10.8 

11.4!  120 

860 

1.6 

1.6 

2.2 

2.9 

3.6 

4.6  5.6 

6.6 

7.6 

8.6 

9.6 

10.5  11.2 

880 

1.2  1.3 

1.5 

1.9 

2.6 

3.3  4.1 

5.2 

6.1 

7.1 

8.2 

9.2  10.1 

900 

1.1 

1.1 

1.2 

1.3 

1.7 

2.2   2.9 

3.8 

4.8 

5.7 

6.8 

7.9!  8.8 

920 

1.1 

1.0 

1.0 

1  i 

1.1 

1.4 

1.9 

2.6 

34 

4.4 

5.3 

6.3 

7.4 

940 

1.5 

1.1 

0.8 

0.9 

1.0 

1.1 

1.3 

1.6 

2.3 

3.1 

3.9 

5.0  5.9 

960 

23 

1.7 

1.3 

0.9 

0.7 

0.8 

0.9 

1.2 

1.4 

2.0 

2.8 

3.5  4.6 

980 

30 

2.5 

1.9 

1.4 

1.2 

1.0  0.8 

0.9 

1.1 

1.4 

1.7 

2.4 

3.3 

1000 

37 

3.2 

2.6 

2.1 

1.7 

1.3  0.9 

0.7 

o.r 

1.0 

1.2 

1.6 

2.2, 

850 

—I 

920  j 

soo 

810 

820  |  830 

840 

860 

870 

880 

890 

900 

910 

40 


TABLE  XXXI. 

Perturbations  by  Mars. 
Arguments  II.  and  IV. 
IV. 


TABLE  XXXII. 

Pert's.  by  Jupiter 
Arg's.  II.  and  V. 
V. 


I'll. 

.... 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

° 

10 

20  ! 

30  ] 

0 

2.2 

3.0 

3.8 

4.8 

5.8 

6.9 

7.8 

8.4 

9.5 

15.3 

15.1 

15.0 

15.0 

20 

1.5 

2.1 

2.6 

3.4 

4.4 

5.5 

6.5 

7.6 

8.7 

14.9 

14.9 

14.7 

14.8 

40 

1.0 

1.4 

1.8 

2.5 

3.2 

4.0 

5.2 

6.0 

7.1 

14.7 

14.6 

14.6 

14.5 

60 

1.0 

1.1 

1.3 

1.8 

2.3 

3.0 

3.7 

4.8 

5.8 

14.4 

14.4 

14.4 

14.4 

80 

1.3 

1.1 

1.2 

1.4 

1.6 

2.2 

2.7 

3.6 

4.5 

13.4 

13.9 

14.0 

14.2 

100 

1.7 

1.3 

1.2 

i.2 

1.3 

1.6 

2.0 

2.6 

3.3 

13.2 

13.4 

13.6 

13.7 

120 

2.4 

2.0 

1.5 

1.4 

1.4 

1.4 

1.7 

1.9 

2.4 

12.3 

12.7 

13.0 

13.3 

140 

3.3 

2.8 

2.3 

2.0 

1.7 

1.5 

1.5 

1.8 

2.1i 

11.3 

11.8 

12.1 

12.5 

160 

4.2 

3.6 

3.1 

2.6 

2.1 

2.0 

1.7 

1.7 

1.9 

10.2 

10.7 

11.2 

11.7 

180 

5.2 

4.6 

4.0 

3.5 

3.1 

2.5 

2.0 

2.0 

1.9 

9.1 

9.6 

10.1 

10.7 

200 

6.1 

5.5 

5.0 

4.4 

3.9 

3.5 

2.8 

2.7 

2.9 

7.8 

8.3 

8.9 

9.5 

220 

6.7 

6.3 

5.8 

5.4 

4.9 

44 

3.9 

3.2 

3.0 

6.8 

7.2 

7.7 

8.3 

S40 

7.2 

6.9 

6.6 

6.1 

5.6 

5.3 

4.8 

4.2 

3.7 

5.7 

6.2 

6.6 

7.2 

230 

7.6 

7.5 

7.1 

6.8 

6.5 

6.0 

5.6 

5.2 

4.8 

4.8 

5.2 

5.6 

6.1 

280 

7.7 

7.7 

V.5 

V.3 

7.1 

6.7 

6.3 

5.9 

5.5 

3.9 

4.1 

4.7 

5.2 

300 

7.7 

77 

7.7 

7.7 

7.4 

7.2 

7.0 

6.6 

6.1 

3.4 

3.5 

3.9 

4.3 

320 

7.4 

7.4 

7.6 

7.7 

7.6 

7.6 

7.3 

7.1 

6.9 

3.2 

3.1 

3.4 

3.6 

340 

7.2 

7.2 

7.3 

7.5 

7.7 

7.6 

7.6 

7.6 

7.7 

3.2 

3.0 

3.0 

3.1 

360 

7.1 

7.1 

7.1 

7.2 

7.2 

7.6 

7.6 

7.6 

7.5 

3.5 

3.2 

2.9 

2.9 

380 

6.8 

6.9 

7.0 

7.0 

7.0 

7.1 

7.3 

7.5 

7.5 

4.5 

4.0 

,3.4 

3.1 

WO 

6.4 

6.6 

6.6 

6.7 

6.7 

6.9 

7.0 

7.1 

7.3 

5.0 

4.3 

3.8 

3.5 

420 

5.8 

5.9 

6.2 

6.3 

6.6 

6.5 

6.7 

6.7 

6.9 

6.1 

5.2 

4.6 

4.1 

440 

5.5 

5.6 

5.7 

5.8 

6.0 

6.1 

6.3 

6.5 

6.5 

7.5 

6.6 

5.8 

4.9 

460 

5.3 

5.3 

5.4 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

9.0 

7.9 

7.0 

6.3 

480 

5.0 

5.0' 

5.0 

5.1 

5.3 

5.4 

5.5 

5.6 

5.8 

10.5 

9.5 

8.5 

7.6 

500 

4.9 

4.9 

5.0 

5.0 

5.0 

5.1 

5.2 

5.3 

5.3 

12.3 

11.3 

10.0 

9.1 

'•20 

4.8 

4.8 

4.8 

4.8 

4.8 

4.7 

4.9 

5.0 

5.1 

14.0 

12.7 

11.7 

10.7 

540 

4.7 

4.7 

4.6 

4.6 

4.6 

4.5 

4.6 

4.6 

4.7 

15.6 

145 

13.3 

12.3 

560 

4.6 

4.5 

4.5 

4.4 

4.5 

4.5 

4.5 

4.5 

4.4 

17.1 

16.1 

15.1 

14.0 

580 

4.9 

4.7 

4.6 

4.5 

4.4 

4.4 

4.4 

4.4 

4.2 

18.6 

17.4 

16.5 

15.7 

600 

5.3 

4.9 

4.8 

4.7 

4.5 

4.4 

4.4 

4.3 

4.1 

19.8 

19.0 

17.9 

17.0 

620 

5.9 

5.5 

5.1 

4.8 

4.6 

4.5 

4.4 

4.3 

4.2 

20.8 

20.1 

19.2 

18.4 

640 

6.6 

6.1 

5.6 

5.4 

5.0 

4.7 

4.6 

4.5 

4.3 

21.6 

20.9 

20.2 

19.5 

060 

7.5 

6.8 

6.3 

5.9 

5.5 

5.3 

4.9 

4.8 

4.6 

22.1 

21.6 

21.0 

20.4 

680 

8.5 

7.8 

7.3 

6.5 

6.1 

5.6 

5.4 

5.1 

4.8 

22.3 

22,0 

21.6 

21.2 

700 

9.7 

8.9 

8.1 

7.6 

7.0 

6.3 

5.9 

5.6 

5.3 

22.2 

22.0 

21.7 

21.5 

720 

10.8 

10.0 

9.3 

8.5 

7.9 

7.2 

6.0 

6.1 

5.8 

22.0 

21.9 

21.7 

21.6 

740 

11.5 

11.0 

10.2 

9.7 

8.9 

8.2 

7.6 

6.9 

6.5 

21.6 

21.6 

21.5 

21.5 

760 

12.1 

11.8 

11.3 

10.5 

10.0 

9.3 

8.5 

7.9 

7.3 

21.2 

21.1 

21.1 

21.2 

780 

12.8 

12.3 

11.9 

11.4 

10.9 

10.2 

9.6 

9.0 

8.2 

20.4 

20.5 

20.6 

20.7 

800 

12.9 

12.9 

12.5 

12.1 

11.7 

11.2 

10.5 

9.8 

92 

19.6 

19.8 

19.9 

20.1 

820 

12.5 

12.7 

12.8 

12.7 

12.2 

11.9 

11.2 

10.7 

10.1 

18.8 

19.0 

19.2 

19.4 

840 

12.0 

12.4 

12.6 

12.8 

12.6 

12.4 

12.2 

11.5 

10.9 

18.1 

18.2 

18.4 

18.6 

860 

11.2 

11.8 

12.3 

12.5 

12.7 

12.5 

12.5 

12.3 

11.7 

17.4 

17.5 

17.6 

17.9 

880 

10.1 

11.0 

11.5 

12.1 

12.3 

12.6 

12.6 

12.4 

12.3 

16.9 

16.9 

16.9 

17.1 

900 

8.8 

9.8 

10.6 

11.3 

11.8 

12.2 

12.4 

12.5 

12.4 

16.3 

16.4 

16.4 

16.5 

920 

7.4 

8.4 

9.3 

10.2 

11.0 

11.5 

12.1 

12.2 

12.3 

16.0 

15.9 

15.9 

16.0 

940 

5.9 

7.1 

8.1 

8.9 

9.9 

10.7 

11.2 

11.7 

12.1 

15.8 

15.7 

15.7 

15.6 

960 

4.6 

5.6 

6.7 

7.7 

8.7 

9.4 

10.2 

10.9 

11.4 

15.5 

15.4 

15.3 

15.4; 

980 

3.3 

4.2 

5.2 

6.2 

7.3 

8.2 

8.9 

9.9 

10.6 

15.3 

15.2 

15.2 

15,1 

1000 

2.2 

3.0 

3.8 

4.8 

5.8 

6.9 

7.8 

8.7 

9.5 

15.3 

15.1 

15.0 

15.0 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

0 

10 

~^~ 

30 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


11.   30  I  40  i  50 

60  '  70   80   90   100  110  120  130 

I       : 

„  i  ,,  '  ,,  ~77~  --   -  i  -.  "~~~ 

140 

150 

0  15.0  14.8  14.7 

14.7  14.6  14.5  14.5 

14.4  14.5  14.5 

14.6 

14.7 

14.8 

20  148  14.7  14.6 

14.4  14.4'14.2  14.2 

14.1!  14.1  14.1 

14.1 

14.1 

14.2 

4;)   4.o  144  14.4 

14.3  14.2  14.1  13.9 

13.8  13.8  13.8 

13.8 

13.8 

13.7 

60  i  14.4  14.3  14.3 

14,2 

14.1  ,13.9  13.8 

13.6  l.?5  13.5 

13.5 

13.4 

13.3 

80!  14.2  '  14.2  i  14.1  14.5 

14.0 

13.8  13.7 

13.5  13.4  13.2 

13.1 

13.0  !  13.1 

100  13.7 

13.7;  13.9 

13.9 

13.8 

13.7  13.6 

13.5 

13.4  13.2 

13.0 

12.8  12-.7 

j 

120 

13.3 

134  13.4 

13.5 

13.6  !  13.5  13.5 

13.3 

13.3 

13.2 

13.0 

12.8  12.6 

140 

12.5 

12.8  13.0 

13.1 

13.2 

13.2 

13.3 

13.2 

13.1  13.0  12.9 

12.8  12.6 

160 

11.7 

12.0 

12.4 

12.6 

12.7 

12.8 

12.9 

12.9 

13.0  12.9 

12.8 

12.7 

12.5 

180 

10.7 

11.1 

11.6 

11.9 

12.2 

12.3  12.5 

12.5 

12.6  12.7 

12.8 

12.6 

12.5 

200 

9.5 

10.0 

10.6 

11.0 

11.5 

11.7 

11.9 

12.2 

12.2 

12.3 

12.4 

12.3 

12.3 

220 

8.3 

8.8 

9.5 

9.9 

10.4 

10.8 

11.3  '.11.5 

11.8 

11.9 

12.0 

12.0 

12.0 

240 

7.2 

7.7 

8.2 

8.9 

9.4 

9.8 

10.3 

10.6  11.0 

11.3 

11.5 

11.7 

11.8 

260 

6.1 

6.5 

7.1  1  7.6 

8.3 

8.8 

9.3 

9.7 

10.1 

10.5 

10.9 

ll.O 

11.2. 

280 

5.2 

5.5 

6.0  |  6.5 

7.1 

7.6 

8.2 

8.7 

9.2 

9.6 

10.0 

10.4 

10.61 

300 

4.3 

4.7 

5.1   5.5 

6.1 

6.6 

7.1 

7.6 

8.1 

8.7 

9.1 

9.4 

9.9 

320 

3.6 

3.9 

4.3   4.6 

5.1 

5.4 

6.0 

6.6 

7.2 

7.7 

8.1 

8.5 

8.9 

340 

3.1 

3.3 

3.5 

3.8 

4.1 

4.5 

5.0 

5.4 

6.1 

6.6 

7.2 

7.6 

8.0 

360 

2.9 

3.0 

3.1   3.3 

3.6 

3.8 

4.1 

4.5 

5.0 

5.5 

6.1 

6.6 

7.1 

380 

3.1 

2.8 

2.8   2.7 

2.8 

2.9 

3.0 

3.2 

3.5 

4.1 

4.6 

5.0 

5.6 

400 

35 

3.1 

2.9  j  2.9 

2.8 

2.8 

3.0 

3.1 

3.4 

3.8 

4.2 

4.7 

5.2 

420 

4.1 

3.6 

3.3  3.1 

2.8 

2.7 

2.8 

2.9 

3.1 

3.2 

3.5 

3.8 

4.3 

440 

4.9 

4.4 

3.9   34 

3.1 

2.7 

2.8 

2.7 

2.8 

3.1 

3.1 

3.2 

3.5 

460 

6.3 

5.4  i  4.8 

4.3 

3.7 

3.2 

2.9 

2.8 

2.8 

2.7 

2.7 

2.8 

3.2 

480 

7.6 

6.7'  59   5.2 

4.6 

4.1 

3.6 

3.1 

30 

2.8 

2.8 

2.6 

2.7 

500 

9.1 

8.1 

7.2   6.4 

5.7 

5.0 

4.4 

3.9 

3.4 

3.2 

3.1 

2.9 

2.7 

1 

520 

10.7 

9.5 

8.7  i  7.7 

6.9 

6.1 

5.5 

4.8 

4.2 

3.8 

3.5 

3.2 

3.1 

540  12.3 

11.  l|  10.  21  9.1 

8.4 

7.4 

6.6 

5.9 

5.3 

4.7 

4.1 

3.8 

3.5 

560  j  14.0 

13.0  11.9,  10.8 

9.9 

8.7 

7.9 

7.1 

6.4 

5.8 

5.2 

4.5 

4.1 

580j  15.7  14.5 

13.  6  i  12.5 

11.4 

10.4 

9.3 

8.3 

7.7 

6.9 

6.2 

5.5 

5.0 

600  17.0  |  16.0  15.0  i  14.0 

13.1 

12.0 

11.0 

10.1 

9.2 

8.2 

7.5 

6.7 

6.0 

620 

18.4 

17.4  16.  5  !  15.5 

14.7 

13.6 

12.6 

11.6 

10.7 

98 

9.0 

8.0 

7.3 

640 

195 

18.5  1  7.9  I  17.0 

16.0 

15.1 

14.2 

13.1 

12.2 

V.3 

10.8 

9.4 

8.7 

660  !  20.4 

ra.7 

IS.  9  1S.1 

17.4 

16.3 

15.6 

146 

13.7 

12.8 

11.9 

ll.O  10.1 

680  j  21.  2 

20.5  19.9,  19.1 

18.5 

17.6 

16.8 

16.0 

15.1 

14.2 

13.5 

12.5  11.6 

700  21.5  21.0  20.6  200 

19.3 

18.7 

18.0 

17.1 

16.5  i  15.6 

14.7 

13.8  13.0 

720 

21.6  21.2  :21.0  20.5 

20.0 

19.3 

18.9 

18.3 

17.5  16.8 

16.1 

15.1  1  14.3 

740 

21.5 

21.  2  '21.1  20.8 

20.5 

20.0 

19.4 

18.9 

18.4 

17.7 

17.2 

16.8115.7 

760 

21.2 

21.0  21.0  20.8 

20.7  20.3 

20.0 

194 

19.0 

18.6 

17.9  17.4  16.7 

7SO 

20.7 

20.7  20.7;  20.6 

20.6  20.3 

20.2 

19.8 

19.4 

19.1 

18.7  18.1 

17.6 

8uO  20.1 

20.2  |20.3:20.3 

20.4  20.3 

20.1 

19.9 

19.7 

19.3 

19.1 

18.7 

18.2 

820 

19.4 

19.5 

19.7  19.8 

19.9  19.9 

199 

19.8 

19.8 

19.6 

19.2 

18.9 

18.7 

840 

18.6 

18.8 

18.9  19.0 

19.2  19.3 

194 

19.4 

19.4 

19.4 

19.4  19.0 

18.9 

860 

17.9 

18.0 

18.  3  18.4 

18.6  18.7 

18.8 

18.9 

19.0 

19.1 

19.1  [19.0 

18.8 

880 

17.1 

17.2 

17.5  17.6 

17.9  18.0 

18.2 

18.3 

18.5 

18.6 

18.6  i  18.6 

18.7 

900 

16.5 

16.6 

16.8 

1G.9 

17.1  17.1 

17.4 

17.5 

17.7 

17.9 

18.1 

18.2 

18.2 

920 

16.0 

16.0 

16.1 

16.2 

16.4  16.5 

16.7 

16.8 

17.0 

17.2 

17.4 

17.5 

17.7 

940 

15.6  15.5 

15.6!  15.6 

15.7  15.8 

16.0 

16.1 

163  16.5 

16.8  16.8 

17.1 

960 

15.4  15.3  15.3  15.2 

15.2  15.2 

15.3 

15.4 

15.6 

15.7 

15  9  i  16.0 

16.3 

980 

15.1 

15.0  15.0  14.9 

14.9  14.8 

14.9 

14.9 

14.9  15.0 

15.2  !  15.3 

15.5 

1000 

15.0  14.8  |  14.7 

14.7 

14.6  14.5 

14.5 

14.4 

14.5 

14.5 

14.6 

14.7 

14.8 

30 

40   50  j  60  i  70   80  i  90   100  |  110 

120 

130 

140 

150 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 
Arguments  II.  and   V. 
V. 


II. 

150 

160  |  170 

180 

190 

200  |  210 

220 

230 

240 

250 

260 

270 

0 

14.8 

15.0 

15.3 

15.5 

15.8 

15.9 

16.2 

16.3 

16.7 

17.0 

17.1 

17.3 

17.5 

20 

14.2 

14.3 

14.6 

H.8 

14.9 

15.2  15.5 

15.7 

15.9 

16.2 

16.6 

16.8 

17.1 

40 

13.7  13.7 

13.9 

14.1 

14.3 

14.5  14.8 

15.0 

15.3 

15.5 

15.8 

16.2 

16.4 

60 

13.3 

13.2 

13.4 

13.5  13.6 

13.8  14.1 

14.3 

14.6 

14.8 

15.1 

15.5 

15.8 

80 

13.1 

13.0 

13.0 

13.0  13.1 

13.1  13.3 

13.5 

13.8 

14.1 

14.4 

14.5 

15.1 

100 

12.7 

12.7 

12.7 

12.6 

12.7 

12.6 

12.8 

12.9 

13.1 

13.4 

13.7 

14.0 

14.2 

120 

12.6 

12.5 

12.5 

12.4 

12.3 

12.2 

12.3 

12.3 

12.6 

12.S 

13.0 

13.3 

13.6 

140 

12.6J  12.4 

12.4 

12.3 

12.1 

12.0 

12.0 

12.0 

12.1 

12.1 

12.3 

12.5 

12.8 

160 

12.5 

12.3 

12.2 

12.1 

12.1 

11.9 

11.8 

11.8 

11.8 

11.8 

11.9 

12.0 

12.2 

180 

12.5 

12.3 

12.2 

12.1 

11.9 

11.8 

11.7 

11.5 

11.5 

11.5 

11.6 

11.7 

11.8 

200 

.12.3 

12.2 

12.2 

12.0 

11.9 

11.7 

11.7 

11.5 

11.4 

11.3 

11.2 

11.3 

11.5 

220 

12.0 

12.0 

12.1 

12.0 

11.8 

11.6 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

240 

11.8 

11.8 

11.9 

11.9 

11.8 

11.6 

11.5 

11.4 

11.3 

11.2 

11.1 

11.1 

11.0 

2RO 

11.2  111.  5 

11.6 

11.6  11.6 

11.5 

11.3 

11.3 

11.3 

11.2 

11.1 

11.0 

10.9 

280  1  10.6 

10.8 

11.1 

11.2 

11.2 

11.2 

11.3  11.3 

11.2 

11.2 

11.1 

11.0 

IP  9 

300 

9.9 

10.1 

10.5 

10.8 

10.9 

11.0 

11.  1 

11.0 

11.0 

11.0 

11.0 

11.1 

10.9 

320 

8.9 

9.4 

9.7 

10.1 

10.4 

10.5 

10.7  i  10.8 

10.8 

10.8 

10.8 

10.8 

10.9 

340 

8.0 

8.5 

9.1 

9.3 

9.6 

9.9 

10.2  1  10.3 

10.5 

10.6 

10.6 

10.7 

10.7 

360 

7.1 

7.5 

8.0 

8.4 

8.9 

9.2 

9.5 

9.8 

10.1 

10.3 

10.4 

10.5 

10.5 

380 

5.6 

6.2 

6.8 

7.J 

7.8 

8.3 

8.9 

9.3 

9.7 

10.0 

10.0 

10.1 

10.2 

400 

5.2 

5.6 

6.2 

6.6 

7.0 

7.5 

7.9 

8.4 

8.8 

9.1 

9.4 

9.7 

9.9 

420 

4.3 

4.8 

5.3 

5.8 

6.2 

6.6 

7.1 

7.4 

7.9 

8.4 

8.7 

9.1 

9.4 

440 

3.5 

3.9 

4.4 

4.9 

5.4. 

5.7 

6.2 

6.7 

7.1 

7.6 

7.9 

8.4 

8.7 

460 

3.2 

3.3 

3.8 

4.1 

4.5 

4.9 

5.4 

5.7 

6.3 

6.7 

7.2 

7.7 

8.0 

480 

2.7 

2.9 

3.2 

3.6 

3.9 

4.3 

4.7 

5.0 

5.4 

59 

6.3 

6.8 

7.3 

500 

2.7 

2.7 

2.9 

3.1 

3.4 

3.6 

4.0 

4.4 

4.8 

52 

5.7 

5.9 

6.4 

520 

3.1 

2.8 

2.9 

3.0 

3.1 

3.2 

3.5 

3.8 

4.2 

4.7 

4.9 

5.4 

5.7 

540 

3.5 

3.2 

3.1 

3.0 

30 

3.0 

3.3 

3.5 

3.7 

4.1 

4.3 

4.7 

5.1 

560 

4.1 

3.8 

3.6 

3.3 

3.2 

3.2  !  3.2 

3.3 

3.5  3.7 

4.0 

4.3 

4.5 

580 

5.0 

4.6 

4.2 

4.0 

3.6 

3.5   3.3 

3.2 

3.4 

3.5 

3.7 

4.0 

4.2 

600 

6.0 

5.4 

5.1 

4.6 

4.3 

3.9  3.7 

3.5 

3.5 

3.6 

3.7 

3.8 

4.0 

620 

7.3 

6.6 

6.0 

5.6 

5.1 

4.6   4.2 

4.0 

3.9 

3.8 

3.9 

3.9 

4.0 

640 

8.7 

7.8 

T.3 

6.6 

6.1 

5.5 

5.2 

4.7 

4.4- 

4.2 

4.0 

4.0 

4.1 

660 

10.1 

9.3  1  8.6 

7.7 

7.2 

6.5 

6.2 

5.9 

5.3 

4.9 

4.6 

4.5 

4.4 

680 

11.6 

10.8  10.0 

9.3 

8.5 

7.5 

7.3 

6.7 

6.3 

5.8 

5.5 

5.2 

4.9 

700 

13.0 

12.1  1  11.5 

10.7 

9.9 

9.0 

8.5 

7.8 

7.4 

6.9 

6.3 

6.0 

5.8 

720 

14.3 

13.5 

12.8 

12.1 

11.3 

10.6 

9.8 

9.1 

8.7 

8.0 

7.6 

7.0 

6.6 

740 

15.7 

14.9 

14.2 

13.4 

12.7 

12.0 

11.2 

10.5 

9.7 

9.3 

8.9 

8.2 

7.7 

760 

16.7 

15.9 

15.5 

14.7 

13.9 

13.3 

12.6 

11.8 

11.2 

10.5 

10.0 

9.5 

9.0 

780 

17.6 

17.0 

16.4 

15.7 

15.1 

14.6 

13.8 

13.2  12.6 

11.9 

11.2 

10.S 

10.  ;2 

800 

18.2 

17.8 

17.3 

16.8 

16.2 

16.0 

15.0 

14.3 

13.7 

13.1 

12.6 

12.0 

11.5 

820 

18.7 

18.3 

18.0 

17.6 

17.0 

16.6 

16.0 

153 

14.9 

14.3 

13.7 

13.1 

12.6  i 

840 

18.9 

18.7 

18.4 

18.2 

17.7 

17.2 

16.8 

16.3 

15.8 

15.3 

14.9 

14.4 

13.8 

860 

18.8 

18.7 

18.6 

18.4 

18.3 

17.9 

17.4 

17.1 

16.7 

16.3 

15.9 

15.4 

15.0 

880 

18.7 

1S.5 

18.6 

18.5 

18.3 

18.2 

18.0 

17.7 

17.4 

17.1 

16.6 

.6.3 

15.9 

900 

18.2 

18.2 

18.3 

18.3 

18.3 

18.1 

18.1 

18.0 

17.8 

17.6 

17.3 

17.0 

16.7 

920 

17.7 

17.9 

18.0 

18.0 

18.1 

18.1 

18.0 

18.  OJ  18.0 

17.8 

17.7 

17.6 

17.3 

940 

17.1 

17.1 

17.4 

17.6 

17.6 

17.7 

17.8 

17.8 

17.9 

18.0 

17.8 

17.8 

17.7 

960 

16.3 

16.5 

16.8 

16.9 

17.1 

17.2 

17.4 

17.5 

17.6117.8 

17.9 

18.0 

17.9 

980 

15.5 

15.7 

16.1 

16.3 

16.5 

16.7 

16.8 

17.0 

17.2 

17.3 

17.6 

17.7 

17.9 

1000 

14.8 

15.0 

15.3 

15.5 

15.8 

15.9 

16.2 

16.3 

16.7 

17.0 

17.1 

17.3 

17.5 

150 

160  170 

180 

190 

200 

210 

220 

230  ,  240 

250 

260 

270 

TAELE   XXXII. 


Perturbations  produced  by  Jupiter. 
Arguments  II.  and  V 
V 


Jl. 

279  280  290  300 

310 

320 

330  340  350 

360  370  ;  380  i  390 

0 

.  i  . 

17.6 

17.5 

„ 
17.5 

17.5!  17.5  |  17.7  !  17.8 

17.9 

17.9 

18.0 

18.0 

17.9 

17.7 

20 

17.1  17.3  17.5;  17.6 

17.8 

17.8 

18.0 

18.1  18.1 

18.1 

18.0 

18.0 

18.0 

40 

16.4  16.8  16.9  17.2 

17.6 

17.7 

17.9 

18.1  18.3 

18.3 

18.4 

18.4 

18.6 

60 

15.8  16.0  16.4  16.7 

16.9 

17.3 

17.6 

17.9  18.2 

18.3 

18.5 

18.5 

18.7 

80 

15.1  15.4  15.7  16.1 

16.4 

16.7 

17.0 

17.5  j  17.8 

18.0 

18.3 

18.5 

18.8. 

100 

14.2 

14.6  15.1  15.0 

15.8 

16.1 

16.5 

17.0 

17.2 

17.5 

17.9 

18.3 

18.7 

120 

13.6 

13.7  14.2 

14.5 

15.0 

15.4 

15.8 

16.2 

16.7 

17.1 

17.3 

17.9 

18.3 

140 

12.8 

13.1  i  13.3 

13.7 

14.2 

14.4 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.7 

160 

12.2 

12.4  j  12.6 

12.9 

13.4 

13.8 

14.1 

14.6 

15.2 

15.5 

16.0 

16.5 

17.1 

180 

11.8 

11.9 

12.1 

12.3 

12.5 

12.8 

13.3 

13.7 

14.4 

14.7 

15.2 

15.7 

16.3 

200 

11.5 

11.5 

11.6 

11.7 

12.0 

12.1 

12.5 

13.0 

13.4 

13.8 

14.3 

14.7 

15.5 

220 

11.1 

11.1 

11.2 

11.3 

11.6 

11.7 

11.9 

12.3 

12.7 

13.0 

13.5 

14.0 

14.5 

240 

11.0 

10.9 

10.9 

11.0 

11.2 

11.3 

11.5 

11.8 

12.1 

12.3 

12.8 

13.2 

13.8 

260 

10.9 

10.8 

10.8 

10.8 

10.9 

10.9 

11.1 

11.3 

11.4 

11.6 

12.0 

12.3 

13.0 

280 

10.9 

10.8 

10.7 

10.6 

10.7 

10.6 

10.8 

11.0 

11.2 

11.3 

11.5 

11.8 

12.2 

300 

10.9 

10.8 

10.7 

10.6 

10.6 

10.5 

10.6 

10.7 

10.8 

10.9 

11.1 

11.4 

11.8 

320 

10.9 

10.7 

10.7 

10.6 

10.6 

10.5 

10.5 

10.6 

10.7 

10.6 

10.7 

11.0 

11.2 

340 

10.7 

10.7 

10.6 

10.5 

10.5 

10.4 

10.5 

10.5 

10.6 

10.5 

10.6 

10.7 

10.8 

360 

10.5 

10.5 

10.5 

10.5 

10.5 

10.4 

10.4 

10.4 

10.4 

10.3 

10.5 

10.6 

10.8 

380 

10.2 

10.3!  103 

10.3 

10.4 

10.3 

10.4 

10.4 

10.4 

10.3 

10.3 

10.4 

10.6 

400 

9.9 

10.0 

JO.O 

10.2 

10.3 

10.2 

10.2 

10.3 

10.4 

10.3 

10.3 

10.3 

10.5 

420 

9.4 

96 

9.8 

9.9 

10.1 

10.2 

10.1 

10.2 

10.2 

10.2 

10.3 

103 

10.4 

440 

8.7 

9.0 

9.2 

9.4 

9.7 

9.8 

10.0 

10.1 

10.2 

10.1 

10.1 

10.2 

10.4 

460 

8.0 

8.4 

8.6 

8.8 

9.1 

9.3 

9.6 

9.9 

10.1 

10.0 

10.0 

10.2 

10.3 

480 

7.3 

7.6 

7.9 

8.4 

8.7 

8.9 

9.1 

9.4 

9.6 

9.7 

9.8 

10.0 

10.1 

500 

6.4 

6.9 

7.2 

7.6 

8.0 

8.3 

8.6 

8.9 

9.2 

9.4 

9.5 

9.7 

9.9 

520 

5.7 

6.1 

6.6 

6.9 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

9.1 

9.4 

9.7 

540 

5.1 

5.4 

5.8 

6.2 

6.7 

7.0 

7.4 

7.7 

8.0 

8.3 

8.6 

8.9 

9.2 

560 

4.5 

49 

5.1 

5.5 

6.0 

6.3 

6.7 

7.2   7.5 

7.7 

8.0 

8.3 

8.7 

580 

4.2 

4.4 

4.8 

5.0 

5.3 

5.7 

6.1 

6.6 

6.9 

7.1 

7.4 

7.7 

8.1 

600 

4.0 

4.2 

4.3 

4.7 

4.9 

5.2 

5.6 

6.0 

6.3 

6.5 

6.8 

7.2 

7.6 

620 

4.0 

4.0 

4.1 

4.3 

4.7 

4.8 

5.1 

5.5 

'  5.8 

6  1 

6.4 

6.7 

7.0 

640 

4.1 

4,1 

4.2 

4.2 

4.4 

4.6 

4.8 

5.1 

5.4 

5.6 

5.9 

6.3 

6.6 

660 

4.4 

4.3 

4.3 

4.3 

4.5 

4.5 

4.7 

4.9 

5.1 

5.3 

5.5 

5.8 

6.2 

680 

4.9 

4.9 

4.7 

4.6 

4.7 

4.5 

4.6 

4.8 

5.0 

5.1 

5.3 

5.5 

5.8 

700 

5.8 

5.4 

5.2 

5.1 

5.0 

4.9 

4.9 

4.9 

5.1 

5.2 

5.3 

5.4 

5.6 

720 

6.6 

6.2 

5.9 

5.7 

5.6 

5.5 

5.4 

5.3 

5.3 

5.3 

5.3 

5.4 

5.5 

740 

7.7 

7.2 

6.8 

6.5 

6.4 

6.1 

6.0 

5.9 

5.8 

5.7 

5.6 

5.5 

5.7 

760 

9.0 

8.2 

7.9 

7.5 

7.2 

6.9 

6.7 

6.5 

6.3 

6.1 

5.9 

5.9 

6.0 

780 

10.2 

9.7 

9.1 

8.4 

8.2 

7.7 

7.6 

7.4 

7.21 

6.9 

6.6 

6.5 

6.5 

800 

11.5  11.0 

10.4 

9.8 

9.4 

8.7 

8.5 

8.3 

8.0 

7.7 

7.6 

7.3 

7.1 

620 

12.6 

12.1 

11.7 

11.2 

10.6 

10.1 

9.7 

9.2 

9.1 

8.6 

8.3 

8.1 

7.9  1 

840 

13.8 

13.2 

12.8  12.3 

11.9 

11.3 

10.9 

10.5 

10.2 

9.6 

9.4 

9.1 

8.9 

860 

15.0 

14.4  j  13.8 

13.5 

13.1 

12.6  12.1 

11.7  11.2 

10.7 

10.4  i  10.1 

10.0 

880  !  15.9  15.4  15.0  14.4 

14.2 

13.7  13.4 

12  9  12.5 

12.0 

11.5!  11  3 

11.1 

900 

16.7 

16.4  15.9  15.5 

15.2 

14.8  14.4 

14.1  13.7 

13.2 

12.8  124 

12.2 

920 

17.3 

17.1  16.8 

16.5 

16.2 

15.7  15.5 

15.2  !  14.8 

143 

140  136 

13.3 

940  17.7  |  17.5  17.3 

17.1  16.9  i  16.6  16.3 

16.1  16.0 

15.5 

15.0  14.7 

14.5 

960  17.9:  17.8  17.6  17.5  1  17.4  17.2  17.0 

16.9  16.8 

16.4 

16.2  15.8 

15.6 

980  17.9;  17.8  17.8  i  17.8  i  17.8  17.8  17.6117.5  17.3 

17.2 

17.0  16.8 

16.6 

1000 

17.5 

17.7'  17.7  17.8  17.9  17.9  18.0  18.0  17.9 

17.7 

17.6  17.5 

17.5 

27C 

280 

290  300 

310  320  330  340  ;  350 

360 

370  1  380  390 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


i  IL 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

510 

0 

17.5 

17.1 

17.0 

16.7 

16.5 

16.3 

16.1 

15.8 

15.6 

15.1 

14.6 

14.3 

13.9 

20 

18.0 

18.1 

17.7 

17.5 

17.5 

17.2 

17.1 

16.8 

16.7 

16.3 

16.0 

15.6 

15.3 

40 

18.6 

18.6 

18.5 

18.4 

18.3 

18.1 

18.0 

17.8 

17.6 

17.3 

17.2 

16.8 

16.5 

60 

18.7 

18.9 

18.9 

18.9 

18.9 

18.7 

18.8 

18.6 

18.7 

18.4 

18.1 

17.9 

17.7 

80 

18.8 

18.9 

19.2 

19.3 

19.4 

19.3 

19.3 

19.3 

19.3 

19.2 

19.2 

18.9 

18.8 

100 

18.7 

18.9 

19.1 

19.4 

19.7 

19.8 

19.8 

19.8 

19.8 

19.8 

19.9 

19.7 

19.7 

120 

18.3 

18.6 

18.9 

19.2 

19.5 

19.8 

20.0 

20.1 

20.3 

20.3 

204 

20.4 

20.4 

140 

17.7 

18.2 

18.6 

18.9 

19.2 

19.6 

20.0 

20.3 

20.5 

20.6 

20.7 

20.8 

21.0 

160 

17.1 

17.6 

17.9 

18.5 

19.0 

19.3 

19.8 

20.2 

20.5 

20.6 

20.9 

21.1 

21.2 

180 

16.3 

16.8 

17.3 

17.9 

18.3 

18.8 

19.3 

19.8 

20.3 

20.6 

20.9 

21.1 

21.4 

200 

15.5 

16.0 

16.5 

17.1 

'17.7 

18.2 

18.6 

19.1 

19.8 

20.2 

20.7 

21.0 

21.4 

220 

14.5 

15.0 

15.6 

16.1 

16.9 

17.4 

18.0 

18.6 

19.0 

19.7 

20.3 

20.7 

21.1 

240 

13.8 

14.2 

14.7 

15.2 

15.9 

16.5 

17.1 

17.7 

18.4 

18.9 

19.5 

20.1 

20.7 

260 

13.0 

13.4 

13.9 

14.4 

15.0 

15.5 

16.3 

16.9 

17.5 

18.0 

18.6 

19.3 

20.0 

280 

12.2 

12.7 

13.0 

13.5 

14.2 

14.7 

15.3 

15.9 

16.7 

17.2 

17.8 

18.4 

19.1 

300 

11.8 

11.9 

12.4 

12.8 

13.3 

13.8 

14.4 

14,9 

15.7 

16.3 

17.0 

17.6 

18.2 

320 

11.2 

11.5 

11.8 

12.2 

12.7 

13.0 

13.6 

14.1 

14.7 

15.3 

16.0 

16.6 

17.4 

340 

10.8 

11.2 

11.4 

11.6 

12.1 

12.4 

12.9 

13.4 

13.9 

14.4 

15.1 

15.7 

16.4 

360 

10.8 

10.8 

11.0 

11.2 

11.6 

11.9 

12.3 

12.6 

13.2 

13.6 

14.2 

14.8 

15.5 

380 

10.6 

10.6 

10.7 

10.9 

11.2 

11.4 

11.9 

12.2 

12.6 

12.9 

13.5 

13.9 

14.5 

400 

10.5 

10.5 

10.6 

10.6 

10.9 

11.1 

11.4 

11.8 

12.2 

12.5 

12.9 

13.3 

13.8 

420 

10.4 

10.4 

10.5 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

12.4 

12.8 

13.3 

440 

10.4 

10.4 

10.4 

10.5 

10.7 

10.8 

10.9 

11.1 

11.3 

11.6 

11.9 

12.2 

12.7 

460 

10.3 

10.4 

10.4 

10.4 

10.6 

10.6 

10.7 

10.9 

11.2 

11.3 

11.7 

11.9 

12.2 

480 

10.1 

10.2 

10.3 

10.4 

10.6 

10.6 

10.7 

10.8 

11.0 

11.2 

11.4 

11.7 

12.0 

500 

9.9 

10.0 

10.1 

10.2 

10.4 

10.5 

10.7 

10.8 

10.9 

11.0 

11.2 

11.3 

11.7 

520 

9.7 

9.8 

9.8 

10.0 

10.2 

10.3 

10.5 

10.6 

10.9 

10.8 

11.1 

11.3 

11.5 

540 

9.2 

9.4 

9.6 

9.8 

10.0 

10.2 

10.3 

10.4 

10.6 

10.7 

10.9 

11.1 

11.4 

560 

8.7 

8.9 

9.1 

9.3 

9.7 

9.8 

10.1 

10.3 

10.5 

10.6 

10.7 

10.8 

11.2 

580 

8.1 

8.5 

8.7 

8.7 

9.2 

9.4 

9.7 

9.9 

10.2 

10.4 

10.6 

10.7 

10.9 

600 

7.6 

7.9 

8.2 

8.5 

8.8 

9.0 

9.3 

9.5 

9.8 

10.0 

10.3 

10.5 

10.7 

620 

7.0 

7.3 

7.6 

7.9 

8.2 

8.5 

8.8 

9.0 

9.4 

9.6 

10.0 

10.1 

10.4 

640 

6.6 

6.8 

7.1 

7.4 

7.7 

7.9 

8.2 

8.6 

8.9 

9.1 

9.4 

9.7 

10.1 

660 

6.2 

6.4 

6.6 

6.9 

7.3 

7.6 

7.9 

8.1 

8.3 

8.6 

8.9 

9.2 

9.5 

680 

5.8 

6.1 

6.2 

6.5 

6.8 

7.0 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

9.0 

700 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.9 

7.1 

7.4 

7.6 

7.9 

8.2 

8.5 

720 

5.5 

5.6 

5.7 

5.9 

6.2 

6.3 

6.5 

6.8 

7.1 

7.2 

7.5 

7.7 

8.0 

740 

5.7 

5.7 

5.7 

5.8 

6.0 

6.1 

6.2 

6.4 

6.7 

6.9 

7.1 

7.2 

7.5 

760 

6.0 

6.0 

6.0 

6.0 

6.0 

6.1 

6.2 

6.3 

6.4 

6.5 

6.7 

6.8 

7.1 

780 

6.5 

6.3 

6.2 

6.2 

6.3 

6.3 

6.3 

6.3 

6.4 

6.4 

6.5 

6.7 

6.8 

800 

7.1 

7.0 

6.7 

6.6 

6.7 

6.5 

6.5 

6.4 

6.5 

6.5 

6.5 

6.6 

8.7 

820 

7.9 

7.6 

7.5 

7.3 

7.2 

7.0 

7.0 

6.8 

6.8 

6.7 

6.6 

6.6 

6.7 

840 

8.9 

8.6 

8.3 

8.1 

7.8 

7.7 

7.6 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

860 

10.0 

9.7 

9.3 

9.0 

8.7 

8.4 

8.2 

8.1 

7.9 

7.7 

7.6 

7.3 

7.2 

880 

11.1 

10.5 

10.4 

10.0 

9.7 

9.5 

9.2 

8.9 

8.7 

8.4 

8.2 

7.9 

7.7 

900 

12.2 

11.8 

11.5 

11.0 

10.8 

10.5 

10.3 

9.9 

9.7 

9.4 

9.0 

8.8 

8.5 

920 

133 

13.0 

12.6 

12.3 

12.1 

11.5 

11.3 

11.0 

10.6 

10.2 

10.1 

9.7 

9.4 

940 

14.5 

14.1 

13.8 

13.5 

13.2 

12.8 

12.5 

11.9 

11.8 

11.3 

11.0 

10.7 

10.4 

960 

15.6 

15.3 

14.9 

14.6 

14.4 

14.0 

13.7 

13.3 

13.0 

12.5 

12.1 

11.8 

11.5 

980 

16.6 

16.3 

16.0 

15.7 

15.6 

15.2 

14.9 

14.6 

14.2 

13.8 

13.6 

12.9 

12.7 

1000 

17.5 

17.1 

17.0 

16.7 

16.5 

16.3 

16.1 

15.8 

15.6 

15.1 

14.6 

14.3 

13.9 

390 

400 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

510 

L  .,..- 

TABLE  XXXII. 

Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


4.c 


!  II. 

510  520 

530 

540 

~ 

550  |  560  570 

580 

590 

600 

610 

620 

630, 

~~ 

'/ 

- 

0 

13.9  13.4 

13.1 

12.7 

12.1  11.8  11.3  10.8 

10.2 

9.9 

9.4 

8.9 

8.4 

20 

15.3,  14.9 

14.4 

13.9 

13.5  13.1  12.5 

12.1 

11.5 

11.0 

10.4 

10.0 

9.4 

40 

16.5  16.3 

15.7 

15.4 

15.0  14.3  13.8 

13.4 

12.8 

12.3 

11.7 

11.1 

10.5 

60 

17.7  17.3 

17.0 

16.6 

16.1  15.8  15.3 

14.7 

14.3 

13.7 

13.0 

12.4 

11.8 

80 

18.8  18.5 

18.1 

17.9 

17.4  17.1  16.6 

16.2 

15.7 

15.1 

14.5 

13.9 

13.2 

100 

19.7!  19.5 

19.2 

19.0 

18.8  18.4  17.9 

17.6 

17.0 

16.5 

16.0 

15.2 

14.7 

120 

20.4  20.3 

20.2 

20.0 

19.7  19.5  19.1 

18.8 

18.4 

18.0 

17.3 

16.8 

16.2 

140 

21.0  21.1 

21.0 

20.8 

20.7:20.4  20.2 

19.9 

19.6 

19.3 

18.8 

18.3 

17.7 

160 

21.2 

21.5121.5 

21.6 

21.5,21.3  21.2 

21.0 

20.6 

20.4 

20.1 

19.6 

19.1 

180 

21.1 

21.6121.8 

22.0 

22.0!22.i:21.9 

21.8 

21.6 

21.4 

21.1 

20.7 

203 

200 

21.4 

21.7 

21.9 

22.1 

22.3 

22.5  22.5 

22.5 

22.4 

22.3 

22.1 

21.8 

21.5 

220 

21.1 

21.5 

21.8 

22.2 

22.5 

22.8  23.1 

23.1 

22.9 

22.8 

22.9 

22.6 

22.5' 

240 

20.7 

21.1 

21.5 

21.9 

22.3 

22.7 

23.0 

23.3 

23.4 

23.5 

23.4 

23.3 

23.2 

260 

20.0 

20.6 

21.0 

21.6 

22.0 

22.4 

22.8 

23.2 

23.5 

23.8 

23.8 

23.8 

23.9 

280 

19.1 

19.9 

20.4 

20.9 

21.5 

22.0 

22.4 

23.0 

23.3 

23.7 

24.0 

24.1 

24.1 

300 

18.2 

19.0 

19.6 

20.3 

20.7 

21.3 

21.8 

22.3 

23.0 

23.4 

23.S 

24.1 

24.3 

320 

17.4 

18.9 

18.7 

19.4 

20.0 

20.6 

21.1 

21.8 

223 

22.9 

23.3 

23.7 

24.2 

340 

16.4 

17.0 

17.6 

18.5 

19.2 

19.9 

20.4 

21.1 

21.6 

22.2 

22.8 

23.3 

23.7 

360 

15.5 

16.2 

16.7 

17.4 

18.2 

18.9 

19.5 

20.1 

20.8 

21.5 

22.0 

22.6 

23.2 

380 

14.5 

15.2 

15.9 

16.6 

17.1 

17.9 

18.6 

19.3 

19.8 

20.5 

21.1 

21.8 

22.5 

400 

13.8 

14.4 

14.9 

15.6 

16.2 

16.8 

17.6 

18.4 

19.1 

19.7 

20.3 

20.9 

21.5 

420 

13.3 

13.7 

14.2 

14.8 

15.3 

16.0 

16.5 

17.4 

18.0 

18.7 

19.4 

20.0 

20.6 

440 

12.7 

13.1 

13.6 

14.1 

14.6 

15.2 

15.7 

16.4 

17.1 

17.8 

18.4 

18.9 

19.6 

460 

12.2 

12.7 

130 

13.5 

13.9 

14.4 

15.0 

15.6 

16.1 

16.9 

17.5 

18.2 

18.7, 

480 

12.0 

12.2 

12.5 

13.0 

13.4 

13.9 

14.3 

14.8 

'15.3 

15.9 

16.6  17.3 

17.9 

500 

11.7 

12.0 

12.2 

12.6 

12.9 

13.3 

13.8 

14.3 

14.7 

15.2 

15.7 

16.4 

16.9 

520  11.5 

11.9 

12.0 

12.3 

12.6 

13.0 

13.2 

13.8 

14.2 

14.7 

15.1 

15.5 

16.2 

540  11.4 

11.6 

11.9 

12.2 

12.4 

12.7 

12.9 

13.3 

13.7 

14.2 

14.6 

15.0 

154 

5RO  11.2 

11.4 

11.5 

11.9 

12.1 

12.4 

12.7 

13.1 

13.4 

13.8 

14.1 

14.5 

14.9 

580  10.9 

11.2 

11.4 

11.6 

11.9 

12.2 

12.4 

12.8 

13.1 

13.5 

13.8 

14.2 

14.5 

600  10.7 

10.8 

11.  1 

11.5 

11.7 

12.0 

12.2 

12.5 

12.8 

13.1 

13.4 

13.8 

14.2 

620  10.4 

10.7 

10.7 

11.1 

11.4 

11  J 

12.0 

12.3 

12.5 

12.9 

13.1 

13.4 

13.8 

640  10.1 

10.4 

10.6 

10.7 

11.0 

11.3 

11.6 

12.0 

12.3 

12.6 

12.9 

13.2 

13.5 

660   9.5 

9.9 

10.2 

10.5 

10.6 

11.0 

11.3 

11.6 

11.9 

12.3 

12.6 

12.9 

13.2 

680   9.0 

9.3 

9.6 

10.0 

10.3 

10.5 

10.8 

11.3 

11.5 

11.9 

12.2 

12.4 

12.8 

700   8.5 

8.9 

9.1 

9.5 

9.8 

10.1 

10.3 

10.7 

11.1 

11.4 

11.8 

12.1 

12.4 

720   8.0 

8.3 

8.5 

9.0 

9.2 

9.6 

9.9 

10.2 

10.5 

10.9 

11.3 

11.7 

12.0 

740   7.5 

7.8 

8.0 

8.3 

8.6 

9.0 

9.3 

9.7 

9.9 

10.4 

10.8 

11.1 

11.5 

760 

7.1 

73 

7.5 

7.9 

8.1 

8.4 

8.6 

9.1 

9.4 

9.7 

10.1 

10.5 

10.9 

780 

6.8 

7.0 

7.1 

7.3 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.4 

9.8 

10.2 

800 

6.7 

6.8 

6.8 

7.0 

7.1 

7.3 

7.5 

7.8 

8.2 

8.5 

8.8 

9.1 

9.5 

820 

6.7 

6.8 

6.6 

6.8 

6.9 

7.0 

7.1 

7.4 

7.6 

7.9 

8.1 

8.4 

8.7 

840 

6.8 

6.8 

6.8 

6.8 

6.8 

6.9 

6.9 

7.1 

7.2 

7.4 

7.6 

7.9 

8.1 

SCO 

7.2 

7.1 

7.1 

7.0 

6.9 

6.9 

6.8  1  6.8 

6.9 

7.1 

7.2 

7.3 

7.6 

880 

7.7 

7.5 

7.4 

7.3 

7.1 

7.0 

6.8 

6.8 

6.7 

6.8 

6.8!  7.0 

7.2 

900 

8.5 

8.2 

7.9 

7.7 

7.5 

7.3 

7.2 

7.1 

6.9 

6.9 

6.8 

6.8 

6.8 

920 

9.4 

9.2 

8.7 

8.4 

8.1 

7.9 

7.6 

7.4 

7.1 

7.0 

6.9 

6.8 

6.7 

940 

10.4 

10.0 

9.7 

9.4 

8.9 

8.6 

8.3 

8.1 

7.7 

7.4 

7.1  6.9 

6.7 

960 

11.5 

11.2 

10.7 

10.4 

9.8 

9.5 

9.1 

8.8 

8.5 

8.1 

7.7   7.4 

7.1 

980 

12.7 

12.3 

11.8 

11.5 

11.1 

10.6 

10.0 

9.7 

9.2 

8.9 

8.5 

8.1 

7.7 

1000 

13.9 

13.4 

13.1 

12.7 

12.1 

11.8 

11.3 

10.8 

10.2 

9.9 

9.4 

8.9 

8.41 

510 

520 

530 

540 

550 

560 

570  580 

5:)0 

600 

610 

620 

630  1 

46 


TABLE  XXXII. 


Perturbations  produced  by  Jupiier. 

Arguments  II.  and   V. 

V. 


II. 

680 

640 

650 

660  670 

680 

690 

700 

710 

720  730 

740 

750 

0 

8.4 

8.0 

7.7 

7.3  6.9 

6.7 

6.5 

6.5 

6.3 

6.2  6.2 

6.4 

6.5 

20  9.4 

9.0 

8.4 

8.0  7.5 

7.1 

6.9 

6.7 

6.4 

6.3  6.0 

6.1 

6.1 

40  10.5 

10.1 

9.4 

8.9 

8.3 

7.8 

7.4 

7.0 

6.6 

6.4;  6.2 

5.9 

5.8 

60 

11.8 

11.3 

10.6 

10.1 

9.3 

8.7 

8.2 

7.7 

7.2 

6.81  6.4 

6.2 

5.8 

80 

13.2 

12.7 

12.0 

11.3  10.5 

9.9 

9.2 

8.7 

8.1 

7.6  7.1 

6.6 

6.2 

100 

14.7 

14.1 

13.4 

12.8  12.0 

11.3 

;o.e 

9.9 

9.1 

8.5 

7.9 

7.3 

6.8 

120 

16.2 

15.4 

14.9 

14.2  13.4 

12.7 

12.0 

11.3 

10.4 

9.8 

8-9 

8.2 

7.6 

140 

17.7 

17.2 

16.4 

15.6  14.9 

14.2 

13.4 

12.7 

11.9 

11.1  10.2 

9.6 

8.8 

160 

19.1 

18.6 

17.9 

17.3  16.6 

15.7 

15.0 

14.2 

13.3 

12.6  11.7 

10.9 

10.0  j 

180 

203 

19.9 

19.4 

18.8 

18.0 

17.3 

16.7 

15.8 

15.0 

14.1 

13.2 

12.4 

11.5  ! 

200 

21.5 

21.2 

20.8 

20.2 

19.3 

18.9 

18.1 

17.5 

16.6 

15.7 

14.9 

14.0 

13.1 

220 

22.5 

22.3 

21.9 

21.5 

21.0 

20.3 

19.7 

19.0 

18.2 

17.5 

16.6 

15.5 

14.7 

240 

23.2 

23.0 

22.9 

22.5 

22.0 

21.6 

21.1 

20.5 

19.8 

19.1 

18.2 

17.3 

16.4' 

260 

23.9 

23.8 

23.7 

23.5 

23.1 

22.7 

22.3 

21.8 

21.2 

20.6 

19.8 

19.1 

18.1  ! 

280 

24.1 

24.3 

24.2 

24.2 

24.0 

23.7 

23.5 

23.1 

22  4 

21.8 

21.2 

20.5 

19.8 

300 

24.3 

24.5 

24.6 

24.6 

24.5 

24.4 

24.2 

23.9 

23.6 

23.1 

22.5 

21.9 

21.2 

320 

24.2 

24.5 

24.7 

24.9 

24.8 

24.8 

24.8 

24.7 

24.4 

24.1 

23.7 

23.1 

22.5 

340 

23.7 

24.2 

24.5 

24.7 

25.0 

25.2 

25.1 

25.0 

25.0 

24.9 

24.6 

24.1 

23.7 

360 

23.2  23.7 

24.2 

24.5 

24.7 

25.0 

25.1 

25.3 

25.4 

25.3 

25.1 

24.9 

24.5 

380 

22.5 

23.1 

23.6 

24.1 

24.4 

24.7 

25.1 

25.2 

25.4 

25.5 

25.4 

25.3 

25.2 

400 

21.5 

22.3 

22.8 

23.4 

23.9 

24.3 

24.7 

25.1 

25.2 

25.4 

25.6 

25.6 

25.5 

420 

20.6 

21.3 

22.0 

22.6 

23.1 

23.6 

24.1 

24.5 

25.0 

25.2 

25.4 

25.6 

25.7 

440 

19.6 

20.3 

21.0 

21.8 

22.3 

22.9 

23.4 

23.9 

24.3  ;  24.8 

25.0 

25.2 

25.6 

460 

18.7 

19.4 

20.1 

20,7 

21.3 

21.9 

22.6 

23.3 

23.6  24.1 

24.6 

24.8 

25.1' 

480 

17.9 

18.5 

19.1 

19.7 

20.3 

21.0 

21.6 

22.2 

22.8 

23.3 

23.8 

24.3 

24.6 

500 

16.9 

17.6 

18.2 

18.8 

19.3 

19.9  |  20.7 

21.4 

21.9 

22.5 

22.9 

23.4 

23.9 

520 

16.2 

16.8 

17.3 

17.9 

18.4 

19.0|  19.7 

20.4 

21.0 

21.6 

21.1 

22.6 

23.0 

540 

15.4 

16.1 

16.6 

17.2 

17.5 

18.1 

18.7 

19.3 

19.9 

20.5 

21.2 

22.7 

22.2 

560 

14.9 

15.4 

16.0 

16.5 

16.9 

17.3 

17.9 

18.4 

18.9 

19.6 

20.1 

20.7 

21.3 

580 

14.5 

15.0 

15.3 

15.9 

16.3 

16.7 

17.1 

17.6 

18.1 

18.7 

19.3 

19.8 

20.31 

600 

14.2 

14.6 

14.9 

15.3 

15.8 

16.3 

16.6 

17.0 

17.4 

17.9 

18.3 

18.9 

19.4 

620 

13.8 

14.2 

14.6 

14.9 

15.1 

15.7 

16.2 

16.6 

16.9 

17.3 

17.6 

18.0 

185 

640 

13.5 

14.0 

14.2 

14.6 

14.8 

15.1 

15.6 

16.1 

16.5 

16.8 

17.1 

17.5 

17.9 

660 

13.2 

13.5 

13.9 

14.3 

14.6 

14.9 

15.2 

15.6 

15.9 

16.4 

16.6 

17.0 

17.3  ! 

680 

12.8 

13.2 

13.5 

13.9 

14.2 

14.5 

14.9 

15.2 

15.6 

16.0 

16.  2  !  16.5 

16.8 

700 

12.4 

12.9 

13.3 

13.5 

13.8 

14.2 

14.5 

14.9 

15.1 

15.6 

15.9 

16.2 

16.4  ; 

720 

12.0 

12.4 

12.8 

13.2 

135 

13.8 

14.2 

14.5 

14.8 

15.1 

15.5 

15.8 

16.  1 

740 

11.5 

11.9 

12.2 

12.6 

12.9 

13.3 

13.8 

14.2 

14.5 

14.8 

15.1 

15.4 

15.7 

760 

10.9 

11.4 

11.8 

12.2 

12.4 

12.8 

i3.2|  13.7 

14.1 

14.5 

14.7 

15.0 

1541 

780 

10.2 

10.6 

11.2 

11.6 

11.9 

12.4 

12.8 

13.2 

13.5 

13.9 

14.3 

14.6 

14.9 

800 

9.5 

10.0 

10.3 

10.9 

11.3 

11.6 

12.1 

12.6 

12.9 

13.4 

13.8 

14.2 

14.5 

820 

8.7 

9.3 

9.7 

10.0 

10.5 

10.9 

11.4 

11.9 

12.3 

12.8 

13.2 

13.6 

14.0 

840 

8.1 

8.4 

8.8 

9.3 

9.6 

10.1 

10.6  11.1 

11.6 

12.1 

12.5 

13.0 

13.4 

860 

7.6 

7.9 

8.1 

8.5 

8.8 

9.2 

9.7)10.2 

10.7 

11.2 

11.7 

12.1 

12.6 

880 

7.2 

7.4 

7.6 

7.8 

8.1 

8.5 

8.8 

9.4 

9.8 

10.2 

10.7 

11.2 

11.8 

900 

6.8 

7.0 

7.1 

7.3 

7.4 

7.8 

8.2 

8.5 

8.9 

9.4 

9.8 

10.3 

10.8, 

920 

6.7 

6.8 

6.8 

6.9 

7.0 

7.0 

7.4 

7.8 

8.1 

8.6 

8.9 

9.4 

9.9) 

940 

6.7 

6.7 

6.7 

6.8 

6.7 

6.8 

6.8 

7.1 

7.4 

7.7 

8.1 

8.4 

8.9 

960 

7.1 

7.0|  6.8 

6.7 

6.5 

6.5 

6.6 

6.7 

6.8 

7.1 

7.3 

7.7 

8.0 

980 

7.7 

7.4  7.1 

6.9 

6.6 

6.5 

6.4 

6.4 

6.3 

6.5 

6.? 

6.9 

7.3 

1000 

8.4 

8.0  7.7 

7.3 

6.9 

6.7 

6.5 

6.5 

6.3 

6.2 

6.2 

6.4 

6.5 

630  i  640  650  660  670 

680 

690  700 

710 

720 

73  0 

740 

750  j 

TABLE  XXXII. 


Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II.   750  760  770  780  790  800 

810  j  820 

830 

840 

850 

860 

870 

0   6.5 

6.81  7.2 

7.5 

8.0 

8.4 

8.8   9.5 

10.1 

10.5  11.0 

11.6 

12.4 

20   6.1 

6.2  6.5 

6.7 

7.0  7.4 

7.91  8.4 

9.0  9.5  10.0 

10.6 

11.1 

40   5.8 

5.9 

5.9 

6.2 

6.4  6.6 

6.9!  7.4 

7.8  8.2  8.8 

9.5 

10.0 

60   5.8 

5.7 

5.7 

5.7 

5.9 

6.1 

6.2 

6.5 

6.9 

7.2  i  7.7 

8.3 

8.8 

80   6.2 

5.8 

5.7 

5.6 

5.4 

5.6 

5.7 

5.9 

6.1 

6.3!  6.7 

7.3 

7.8 

100   6.8 

6.3 

5.9 

5.6 

5.5 

5.3 

5.3 

5.4 

5.4 

5.6 

5.9 

6.3 

6.8 

120   7.6 

7.4 

6.5 

6.0 

5.7 

5.5 

5.1 

5.2 

5.1 

5.1 

5.2 

5.5 

5.8 

140   8.8 

8.1 

7.4 

6.8 

6.2 

5.8 

5.4 

5.2 

5.0 

4.9 

4.8 

5.0 

5.1 

160  10.0 

9.3 

8.5 

7.8 

7.2 

6.5 

5.9 

5.5 

5.1 

5.9 

4.7 

4.7 

4.7 

180  11.5 

10.6 

9.7 

9.0 

8.2 

7.5 

6.9 

6.3 

5.8 

5.2 

4.8 

4.7 

4.5 

200 

13.1 

12.2 

11.2 

10.4 

9.5 

8.8 

7.9 

7.1 

6.5 

5.9 

5.3 

5.0 

4.7 

220 

14.7 

13.8 

129 

120 

11.1 

102 

93 

8.4 

75 

67 

6.1 

66 

52 

240 

164 

15.3 

14.5 

13.6 

12.6 

11.7 

10.7 

9.8 

8.8 

7.9 

7.0 

6.5 

5.9 

260 

18.1 

17.2 

16.3 

15.3 

14.3 

13.3 

12.2 

11.4 

10.4 

9.4 

8.3 

7.7 

6.9 

280 

19.8 

18.9 

17.9 

170 

16.1 

150 

14.0 

13.0 

11.9  |  10.9 

9.9 

8.9 

8.0 

300 

21.2 

20.4 

19.6 

18.7 

17.7 

16.8 

15.8  j  14.7 

13.7 

12.6 

11.5 

10.5 

9.4 

320 

22.5 

21.9 

21.2 

20.4 

19.4 

18.5 

17.4116.5 

15.5 

14.2 

13.2 

12.3 

11.2 

340 

23.7 

23.0 

22.4 

21.8 

21.1 

20.2 

19.21  183 

17.1 

16.1 

15.0 

13.9 

12.9 

360 

24.5 

24.0 

23.6 

23.0 

22.4|21.6 

20.8 

19.9 

18.9 

17.9 

16.8 

15.9 

14.7 

380 

25.2 

24.9 

24.5 

24.0 

23.5 

22.8 

22.1 

21.4 

20.5 

19.5 

18.5 

17.6 

16.5 

400 

25.5 

25.4 

25.1 

24.8 

24.5 

23.9 

23.4 

22.7 

21.9 

21.0 

20.1 

19.2 

18.2 

420 

25.7 

25.6 

25.5 

25.3 

25.0 

24.5 

24.2 

23.7 

23.2 

22.3 

21.5 

20.7 

198 

440 

25.6 

25.6 

25.7 

25.7 

25.5 

25.3 

24.9 

24.6 

24.1 

23.4 

22.7 

22.0 

21.2 

460 

25.1 

25.3 

25.5 

25.6 

25.8 

25.7 

25.4 

25.2 

24.8 

24.3 

23.7 

23.1 

22.5 

480 

24.6 

24.9 

25.2 

25.4 

25.6 

25.6 

25.5 

25.4 

25.2 

24.9 

24.5 

24.1 

23.5 

SCO 

23.9 

24.2 

24.7 

25.0 

25.3 

25.4 

25.5 

25.5 

25.4 

25.2 

24.9 

24.7 

24.3 

520 

23.0 

23.6 

23.9 

24.3 

24.7 

24.9 

25.2 

25.4 

25.4 

25.3 

25.2 

25.1 

24.8 

540 

22.2 

22.6 

23.2 

23.6 

24.0 

24.4 

24.6 

24.9 

25.1 

25.0 

25.1 

25.1 

250 

560 

21.3 

21.7 

22.2 

22.8 

23.2 

23.7 

24.0 

24.3 

24.6 

24.7 

24.8 

24.9 

24.9 

580 

20.3 

20.8 

21.3 

21.8 

22.3 

22.7 

23.2 

23.7 

23.9 

24.1 

24.4 

24.6 

24.7 

600 

19.4 

19.9 

20.4 

20.8 

21.4 

21.9 

22.2 

22.7 

23.1 

23.4 

23.7 

24.1 

24.3 

620 

18.5 

19.0 

19.5 

20.1 

20.5 

20.9 

21.4 

21.8 

22.2 

22.6 

22.9 

23.3 

23.6 

640 

17.9 

18.3 

18.7 

19.2 

19.7 

20.1 

20.5 

22.0 

21.3 

21.7 

22.1 

22.5 

22.8 

1  660 

17.3 

17.6 

18.1 

18.5 

18.9 

19.4 

19.6 

20.1 

20.5 

20.7 

21.2 

21  7 

22.0 

680 

16.8 

17.1 

17.4 

17.8 

18.2 

18.6 

18.9 

19.4 

19.7 

20.1 

20.4 

207 

21.2 

700 

16.4 

16.7 

16.9 

17.3 

17.7 

18.0 

18.3 

18.7 

18.9 

19.2 

19.6 

20.0 

20.3 

720 

16.1 

16.3 

16.5 

16.9 

17.2 

17.6 

17.8 

18.0 

18.3 

18.5 

18.7 

19  S 

19.5 

•  740 

15.7 

16.0 

16.2 

16.5 

16.7 

17.0 

17.3 

17.6 

17.8 

17.9 

18.1 

185 

18.8 

760 

15.4 

15.7 

16.0 

16.1 

16.4 

16.6 

16.7 

17.2 

17.4 

17.4 

17.8 

180 

18.2 

780 

14.9 

15.3 

15.6 

15.9 

16.1 

16.3 

16.5 

16.7 

16.9 

17.1 

17.3 

17.6 

17.7 

800 

14.5 

14.7 

15.2 

15.5 

15.8 

15.9 

16.2 

16.5 

16.6 

16.8 

16.9 

17.1 

17.3 

820 

14.0 

14.4 

14.7 

15.1 

15.4 

15.7 

15.8 

16.1 

16.3 

16.4 

16.6 

16.9 

17.0 

840 

13.4 

13.7 

14.1 

14.5 

15.1 

15.4 

15.4 

15.8 

15.9 

16.1 

16.2 

16.6 

16.7 

860 

12.6 

13.1 

13.5 

13.9 

14.3  i  14.8 

15.2 

15.5 

15.6 

15.8 

16.0 

16.3 

16.4 

880 

11.8 

12.3 

12.8 

13.3 

13.7!  14.1 

14.5 

15.0 

15.3 

15.4 

15.6 

15.9 

16.1 

900 

10.8 

11.3  11.9 

12.4 

13.0 

13.4 

13.7 

14.2 

14.7 

15.0 

15.2 

15.5 

15.7 

920 

9.9 

10.3 

10.8 

11.4 

12.0 

12.5 

12.9 

13.4 

14.0 

14.3 

14.7 

15.0 

15.3  ' 

940 

8.9 

9.4 

9.9 

10.4 

11.0 

11.6 

12.1 

12.5 

13.0 

13.6 

13.9 

14.4 

14.7 

960 

8.0 

8.3  1  8.8 

91 

10.0  '10.6 

11.1 

11.7  12.2 

12.51  13.1  i  13.7  14.1 

980 

7.3 

7.6 

7.9 

8.4 

8.9  i  9.5 

9.9 

10.5 

11.1 

11.6  |  12.1  i  12.8|13.3 

1000 

6.5 

6.8 

7.2 

7.5 

8.0  1  8.4 

8.8 

9.5 

10.0 

10.5 

11.0  |  11.6  12.4 

750  1  760 

770 

780 

790  800 

810  820  !  830  840  850  [  860 

870  | 

48 


TABLE  XXXII. 

Perturbations  produced  by  Jupiter. 

Arguments  II.  and  V. 

V. 


II.  I  870 

880 

890 

900 

910  920 

930 

940 

950 

960 

970 

980 

990 

1000 

0  !  12.4 

12.9 

13.2 

13.6 

13.9  14.2 

14.4 

14.8 

15.0 

15.1 

15.1 

15.2 

15.2 

15.3 

20  li.l 

11.7 

12.2 

12.7 

13.2  13.6 

13.8 

14.1 

14.4 

14.7 

14.8 

15.0 

14.9 

14.9 

40  10.0 

10.5 

11.1 

11.7 

12.3  12.6 

13.0 

13.4 

13.7 

14.1 

14.3 

14.6 

14.7 

14.7 

60   8.8 

9.4 

9.9 

10.6 

11.2  11.8 

12.1 

12.6 

12.9 

13.3 

13.6 

13.9 

14.2 

14.4 

80   7.8 

8.3 

8.7 

9.3 

10.0  10.5 

11.1 

11.6 

12.1 

12.5 

12.8 

13.2 

13.5 

13.8 

100 

6.8 

7.2 

7.6 

8.1 

8.6  9.4 

9.9 

105 

10.9 

11.4 

12.0 

12.4 

12.8 

13.2 

120 

5.8 

6.1 

6.6 

7.1 

7.6  8.1 

8.7 

9.4 

9.9 

10.4 

10.8 

11.4 

11.8 

12.3 

140 

5.1 

5.3 

5.6 

6.0 

6.5  1  7.0 

7.5 

8.2 

8.7 

9.3 

9.7 

10.3 

10.8 

11.3 

160 

4.7 

4.8 

4.8 

5.2 

5.6  !  5.9 

6.3 

6.8 

7.4 

8.0 

8.6 

9.2 

9.7 

10.2 

180 

4.5 

4.5 

4.4 

4.5 

4.8  1  5.1 

5.4 

5.8 

6.2 

6.9 

7.4 

8.0 

8.4 

9.1 

200 

4.7 

4.5 

4.2 

4.2 

4.2 

4.4 

4.6 

5.0 

5.3 

5.7 

6.3 

6.9 

7.4 

7.8 

220 

5.2 

4.7 

4.3 

4.2 

4.1 

4.1 

4.0 

4.3 

4.5 

4.8 

5.1 

5.7 

6.2 

6.8 

240 

5.9 

5.3 

4.7 

4.3 

4.1 

4.0 

3.8 

3.9 

4.0 

4.2 

4.3 

4.7 

5.2 

5.7 

260 

6.9 

6.1 

5.4 

4.9 

4.4 

4.1 

3.8 

3.7 

3.6 

3.7 

3.8 

4.1 

4.3 

4.9 

280 

8.0 

7.2 

6.3 

5.7 

5.2 

4.6 

4.1 

3.8 

3.5 

3.5 

3.5 

3.6 

3.7 

3.9 

300 

9.4 

8.5 

7.5 

6.8 

6.1 

5.4 

4.7 

4.3 

3.9 

3.6 

3.3 

3.3 

3.3 

3.4 

320 

11.2 

10.1 

9.1 

8.1 

7.3 

6.5 

5.7 

5.0 

4.4 

4.0 

3.6 

3.4 

3.2 

3.2 

340 

12.9 

11.8 

10.7 

9.6 

8.7 

7.7 

6.8 

6.0 

5.2 

4.6 

4.1 

3.7 

3.4 

3.2 

360 

14,7 

13.4 

12.3 

11.1 

10.1 

9.2 

8.3 

7.4 

6.4 

5.7 

4.9 

4.3 

3.8 

3.5 

380 

16.5 

15.4 

14.2 

13.0 

11.8 

10.8 

9.7 

8.7 

7.8 

6.9 

6.1 

5.4 

4.6 

4.1 

400 

18.2 

17.2 

16.0 

14.9 

13.8 

12.4 

11.4 

10.4 

9.3 

8.3 

7.3 

6.4 

5.6 

5.0 

420 

19.8 

18.8 

17.7 

16.7 

15.5 

14.4 

13.1 

11.9 

10.9 

9.8 

8.8 

8.0 

6.9 

6.1 

440 

21.2 

20.3 

19.3 

18.3 

17.3 

16.2 

14.9 

13.8 

12.7 

11.5 

10.5 

9.5 

8.4 

7.5 

460 

22.5 

21.6 

20.6 

19.7 

18.9 

17.9 

16.7 

15.6 

14.3 

13.3 

12.2 

10,9 

10.0 

9.0 

480 

235 

22.7 

220 

21.1 

20.2 

19.3 

18.2 

17.3 

16.2 

15.0 

13.8 

12-8 

11.61-10.5 

500 

24.3 

23.8 

23.0 

22.3 

21.6 

20.7 

19.7 

18.8 

17.8 

16.7 

15.4 

14.5 

13.4 

12.3 

520 

24.8 

24.3 

23.7 

23.2 

22.7 

21.9 

21.1 

20.2 

19.2 

18.3 

17.2 

16.1 

15.0 

14.0 

540 

25.0 

24.8 

24.3 

23.9 

23.4 

22.8 

22  1 

21.3 

20.6 

19.7 

18.7 

17.6 

16.6 

15.6 

560 

24.9 

24.8 

24.7 

24-4 

24.0 

23.6 

22.9 

22.4 

21.6 

20.8 

20.0 

19.1 

18.2 

17.1 

580 

24.7 

24.7 

24.6 

24.5 

24.3. 

23.9 

23.5 

23.1 

22.5 

21.9 

21.1 

20.3 

19.5 

18.6 

600 

24.3 

24.3 

24.3 

24.3 

24.3 

24.1 

23.8 

23.5 

23^0 

22.5 

22.0 

21.4 

20.6 

19.8 

620 

23.6 

23.7 

23.9 

24.0 

24.1 

24.1 

23.9 

23.7 

23.4 

23.1 

22.6 

22.1 

21.4 

20.8 

640 

22.8 

23.1 

23.2 

23.4 

23.6 

23.7 

23.8 

23.7 

23.5 

23.2 

22.9 

22.6 

22.1 

21.6 

660 

22.0 

22.3 

22.5 

22.8 

23.0 

23.2 

23.2 

233 

23.2 

23.1 

23.0 

22.8 

22.5 

22.1 

680 

21.2 

21.5 

21.7 

22.0 

22.3 

22.5 

22.6 

22.8 

22.9 

22.9 

22.8 

22.7 

22.7 

22.3 

700 

20.3 

20.7 

20.9 

21.2 

21.5 

21.7 

21.9 

22  2 

22.3 

22.5 

22.5 

22.5 

22.4 

22.2 

720 

19.5 

19.8 

20.1 

20.4 

20.8 

21.1 

21.2 

21.4 

21.6 

21.8 

21.9 

22.0 

22.0 

22.0 

740 

18.8 

19.0 

19.2 

19.6 

19.9 

20.2 

20.5 

20.7 

20.9 

21.1 

21.2 

21.5 

21.5 

2L6 

760 

18.2  18.5 

18.4 

18.8 

19.1 

1.0.4 

19.6 

19.9 

20.1 

20.3 

20.5 

20.8 

21.0 

21.2 

780 

17.71  17.8 

18.0 

18.1  18.4 

18.7 

18.8 

19.1 

19.3 

19.5 

19.7 

20.0 

20.2 

20.4 

800 

17.3 

17.4 

17.4 

17.7 

17.9 

18.0 

18.1 

18.4 

18.6 

18.9 

18.9 

19.1 

19.4 

19.6 

820 

17.0 

17.2 

17.2 

17.2 

17.4 

17.4 

17.6 

17.8 

17.8 

18.1 

1S.3 

18.5 

18.6 

18.8 

840 

16.7 

16.8 

16.8 

16.9 

17.2 

17.2 

17.1 

17.1 

17.3 

17.4 

17.5 

17.8 

17.9 

18.1 

860 

16.4 

16.5 

16.5 

16.6 

16.8 

16.7 

16.8 

16.9 

16.9 

17.0 

17.0 

17.1 

17.2 

17.4 

880 

16.1 

16.3 

16.3 

16.5 

16.5 

16.5 

16.6 

16.6 

16.6 

16.6 

16.6 

16.7 

16.7 

16.9 

900 

15.7 

15.9 

16.1 

16.2 

16.3 

16.4 

16.3 

16.3 

16.2 

16.2 

16.2 

16.3 

16.3 

16.3 

920 

153 

15.5 

15.6 

15.9 

16.0 

16.1 

16.1 

16.1 

16.0 

16.1 

16.1 

16.1 

16.0 

16.0 

940 

14.7 

15.9 

15.2 

15.4 

15.7 

15.8 

15.8 

16.0 

15.9 

15.9 

15.9 

15.8 

15.7 

15.8 

960 

14.1 

14.3 

14.5 

14.8 

15.2 

15.5 

15.5 

15.7 

15.7 

15.7 

15.6 

15.6 

15.5 

15.5 

980 

133 

12.7 

13.9 

14.2 

145 

14.8 

15.1 

15.3 

154 

15.5 

15.4 

15.4 

15.4 

15.3 

1000 

12.4 

12.9 

13.2 

13.6 

13.9 

14.2 

14.4 

14.8 

15.0 

15.1 

15.1 

15.2 

15.2 

153 

870 

880 

890 

900 

910 

920 

930 

940 

950 

960 

970 

980  990 

1000 

TABLE  XXXIII. 
Perturbations  produced  by  Saturn. 

Arguments  II  and  VII. 
VII. 


II 

0 

100 

200 

300 

400 

500 

600 

700 

800 

900 

1000 

0 

1.2 

1.5 

1.4 

1.0 

0.7 

0.6 

0.5 

0.5 

0.4 

0.8 

1.2 

100 

0.9 

1.2 

1.3 

1.1 

0.9 

0.8 

0.7 

0.7 

0.6 

0.7 

0.9 

200 

0.7 

0.9 

1.0 

1.1 

1.0 

0.9 

O.S 

0.8 

0.9 

0.8 

0.7 

300 

0.9 

0.8 

0.7 

0.8 

0.9 

1.0 

1.0 

1.0 

1.0 

1.0 

0.9 

400 

1.0 

0.9 

0.6 

0.4 

0.6 

0.9 

1.0 

1.1 

1.1 

1.1 

.0 

500 

1.1 

1.0 

0.8 

0.4 

0.2 

0.5 

1.0 

1.3 

1.3 

1.2 

.1 

600 

1.2 

1.1 

0.9 

0.6 

0.2 

0.2 

0.5 

1.1 

1.5 

1.5 

.2 

700 

1.4 

1.1 

1.0 

0.8 

0.4 

0.1 

0.3 

0.8 

1.4 

1.7 

.4 

800 

1.6 

1.3 

1.0 

0.8 

0.6 

0.4 

0.1 

0.3 

1.0 

1.6 

.6 

900 

1.5 

1.4 

1.1 

0.9 

0.7 

0.6 

0.3 

0.2 

0.6 

1.2 

.5 

1000 

1.2 

1.5 

1.4 

1.0 

0.7 

0.6 

0.5 

0.5 

0.4 

0.8 

.2 

Constant,  l."0 


TABLE  XXXIV. 

Variable  Part  of  Sun's  Aberration. 
Argument,  Sun's  Mean  Anomaly. 


0* 

I* 

II* 

III* 

IV* 

V« 

0 

0.0 

0.0 

0.1 

0.3 

0.5 

0.6 

30 

3 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

27 

6 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

24 

9 

0.0 

0.0 

0.2 

0.3 

0.5 

0.6 

21 

12 

0.0 

0.1 

0.2 

0.4 

05 

0.6 

18 

15 

0.0 

0.1 

0.2 

0.4 

0.5 

0.6 

15 

18 

0.0 

0. 

0.2 

0.4 

0.5 

0.6 

12 

21 

0.0 

0. 

03 

0.4 

06 

0.6 

9 

24 

0.0 

0. 

0.3 

0.4 

0.6 

0.6 

6 

27 

0.0 

0. 

0.3 

0.4 

0.6 

0.6 

3 

30 

0.0 

0. 

0.3  I 

0.5 

0.6 

0.6 

0 

XI« 

x» 

IX* 

VIII* 

VII« 

VI» 

Constant,  0."3 


50 


TABLE  XXXY. 
Moon's  Epochs. 


TEAKS. 

1 

2 

3 

4 

5 

6 

T 

8 

9 

10 

11 

12 

13 

1830 

00174 

4541 

4461 

4638 

9885 

0635 

5979 

9921 

7623 

219 

22H 

458 

468 

1831 

00103 

1749 

4127 

9381 

2357 

6432 

7040 

2378 

6487 

826 

587 

177 

940 

1832  B 

00032 

8957 

3793 

4125 

4829 

2229 

8100 

4835 

5351 

432 

948 

897 

413 

Ifc33 

00235 

6816 

449V) 

9156 

7636 

,-  399 

9219 

7683 

4239 

108 

340 

687 

920 

1834 

001  64 

4024 

4164 

3900 

0107 

41960279 

0140 

3103 

715 

701 

406 

393 

1835 

00093 

1232 

3830 

8644 

2579 

9993 

1340 

2598 

1967 

321 

061 

125 

866 

1836  B 

00022 

8441 

3496 

3388 

5051 

5791 

2400 

5055 

0831 

928 

422 

845 

339 

1837 

00224 

6299 

4202 

8419 

7858 

19603518 

7903 

9719 

605 

814 

63* 

846 

1838 

00153 

3508 

3868 

3163 

0329 

7757  4579 

0360 

8583 

211 

175 

354 

319 

1839 

00082 

0716 

3534 

7907 

2801 

3555  5639 

2818 

7447 

818 

536 

074 

792 

1840  B 

00011 

7925 

3199 

2651 

5273 

9352  6700 

5275 

6310 

424 

896 

793 

265 

1841 

00213 

5783 

3906 

7682 

8080 

55227818 

8123 

5199 

101 

288 

583 

772 

1842 

00142 

2991 

3571 

2425 

0551 

1319  8879 

0580 

4062 

707 

649 

302 

245 

1843 

00071 

0200 

3237 

7169 

3023 

711P 

9939 

3038 

2926 

314 

010 

022 

718 

1844  B 

00000 

7408 

2903 

1913 

5495 

2914 

1000 

5495 

1790 

920 

371 

741 

191 

1845 

00203 

5266 

3609 

6944 

8302 

9083 

2118 

8343 

0678 

597 

763 

531 

698 

1846 

00132 

2475 

3275 

1688 

0773 

488(' 

3179 

0800 

9542 

203 

123 

250 

171 

1847 

00061 

9683 

2941 

6432 

3245 

0678 

4239 

3257 

8406 

blO 

484 

970 

644 

1848  B 

99990 

6892 

2606 

1176 

5717 

6475 

5300 

5715 

7270 

416 

845 

689 

117 

1849 

00192 

4750 

3312 

6207 

8524 

2644 

6418 

8563 

6158 

093 

237 

479 

624 

1850 

00121 

1958 

2978 

0951 

0995 

8442 

7479 

1020 

5022 

700 

597 

199 

097 

1851 

00050 

9167 

2644 

5695 

3467 

4239 

8539 

3477 

3885 

306 

958 

918 

570 

1852  B 

99979 

6375 

2310 

0439 

5939 

0036 

9600 

5935 

2749 

91:-; 

319 

637 

043 

1853 

00181 

4233 

3016 

5469 

8746 

6206 

0718 

8782 

1687 

589 

711 

427 

550 

1854 

00110 

1442 

2681 

0213 

1217 

2003 

1778 

1240 

0501 

196 

o72 

147 

023 

1855 

00039 

>650 

2347 

4957 

3689 

7801 

2839 

3697 

9365 

802 

432 

866 

496 

1856B 

99968 

5859 

2013 

9701 

6160 

3598 

3899 

6155 

8229 

409 

793 

586 

969 

1857 

00171 

3717 

2719 

4732 

8968 

9767 

5018 

9002 

7117 

086 

185 

375 

476 

1858 

00100 

0925 

2385 

9476 

1439 

5565 

6078 

1460 

5981 

692 

546 

095 

949 

1859 

00029 

8134 

2051 

4220 

3911 

1362 

7139 

3917 

4845 

299 

9<)7 

814 

422 

1860  B 

99958 

5342 

1716 

8964 

6383 

7159 

8199 

6374 

3709 

905 

267 

534 

895 

1861 

00160 

3200 

2423 

3995 

9190 

3329 

9317 

9222 

2597 

581 

659 

328 

402 

1862 

00089 

0409 

2088 

8739 

1661 

9126 

0378 

1679 

1461 

188 

020 

043 

875 

1863 

00018 

7617 

1754 

3483 

4133 

4923 

1438 

4107 

0324 

795 

381 

76v 

348 

1864  B 

99947 

4826 

1420 

8227 

6605 

0721 

2499 

6594 

9188 

401 

742 

482 

821 

1865 

00149 

2684 

2126 

3257 

9412 

6890 

3617 

9442 

8076 

07  b 

134 

272 

328 

1866 

00078 

9893 

1792 

8001 

1883 

2687 

4678 

1899 

6940 

685 

494 

991 

801 

1867 

00007 

7101 

1457 

2745 

4355 

8485 

5738 

4357 

5804 

291 

855 

711 

274 

1868  B 

99936 

4309 

1123 

7489 

t.827 

4282 

6799 

6814 

4668 

898 

216 

431 

747 

1869 

00138 

2168 

1829 

2520 

9634 

0452 

7917 

9662 

355H 

574 

608 

220 

254 

1870 

00067 

9376 

1495 

7264 

2105 

6249 

8978 

2119 

2420 

181 

968 

940 

727 

1871 

99996 

6585 

1161 

2008 

4577 

2046 

0038 

4576 

1283 

787 

329 

659 

200 

1872  B 

99925 

3793 

0827 

6752 

7049 

7843, 

1099 

7034 

0147 

394 

690 

378 

673 

1873 

00127 

1651 

1533 

1782 

9856 

4013 

2217 

9881 

9035 

070 

082 

168 

189 

1874 

00056 

9860 

1198 

6526 

2327 

981<i 

3277 

2339 

7899 

677 

448 

888 

653 

1875 

99985 

6068 

0864 

1270 

4799 

5608 

4338 

4796 

6763 

283 

803 

607 

126 

1876  B 

99914 

3277 

0530 

6014 

7280 

1405 

5398 

7254 

5627 

890 

164 

327 

599 

1877 

00117 

1185 

1236 

1045 

0078 

7574 

6517 

0101 

4515 

567 

556 

116 

106 

1878 

00046 

8343 

0902 

5789 

2549 

3372 

7577 

2559 

3379 

173 

917 

836 

579 

1879 

99975 

5552 

0568 

0533 

5021 

9169 

8638 

5016 

2243 

780 

278 

555 

052 

1880  B 

99904 

2760 

0233 

5277 

7493 

4966 

9698 

7473 

1107 

386 

638 

275 

525 

1881 

10106 

0618 

0940 

0308 

0300 

1136 

0816 

0321 

9995 

062 

030 

064 

032 

1882 

00085 

7827 

0605 

5052 

2771 

6933 

1877 

2798 

8859 

669 

891 

784 

505 

1883 

99964 

5035 

0271 

9796 

5243 

2730 

2937 

5236 

7722 

276 

752 

503 

978 

1884  B 

99893 

2244 

9937 

4540 

7715 

8528 

3998 

7693 

6586 

882 

113 

223 

451 

1885 

00095 

0042 

0643 

9570 

0522 

4697 

5116 

0541 

5474 

559  505 

013 

957 

TABLE  XXXV. 
Moon's  Epochs. 


51 


Tears. 

14 

15  1  16 

17  18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

1830 

921 

392 

230 

588  462 

523^536 

52 

60 

44 

04 

51 

47 

08 

99 

'.'9 

89 

52 

1831 

115 

532 

589 

940  937 

29«  703 

:-;o 

70 

41 

65 

53 

94 

48 

2424 

51 

44 

1832  B. 

309 

673J949 

293412070870 

•  17 

81 

38 

36 

55 

42 

'.'7 

48 

40 

14 

36 

1833 

602 

844 

345 

688  9  1  H  84o  037 

88 

02 

45 

07 

61 

92 

53 

77 

77 

77 

27 

1834 

796 

984 

704 

040  38;>  r,  i  :t 

203 

62 

1(3 

42 

77 

68 

40 

03 

01 

01 

30 

18 

1835 

989 

124 

063 

393  St',3 

392 

30 

13 

38 

4.8 

60 

87 

51 

26 

26 

()2 

10 

1836  B. 

183 

265 

423 

745  338 

100  :37 

1  7 

24 

85 

1$ 

67 

34 

01 

50 

51 

64 

01 

1837 

476 

436 

819 

14"  84" 

!H2  704 

94 

35 

42 

ltd 

73 

85 

58 

70 

70 

27 

S3 

1838 

670 

576 

178 

492  •.!•') 

715  870 

72 

46 

38 

00 

75 

3207 

04 

04 

89 

84 

1839- 

864 

7  1" 

537 

845  790 

4*9  037 

4:1 

66 

35 

31 

77 

80 

5*', 

28 

28 

52 

76 

1840  B. 

05s 

857 

897 

197  -65 

J62  204 

26 

67 

32 

02 

70 

27 

"6 

53 

53 

14 

67 

1841 

351 

028 

293 

5^2  766 

"38 

371 

1)4 

78 

39 

73 

85 

77 

82 

81 

si 

77 

59 

1842 

544 

168 

6-V2 

1(44  241 

811 

537 

81 

89 

35 

48 

87 

25 

12 

06 

06 

40 

51 

1843 

738 

308 

012 

297  716 

585  704 

58 

•..it 

32 

14 

80 

12 

61 

30 

81 

"2 

42 

1844  B. 

932 

449 

371 

6^9  191 

36 

10 

20 

85 

01 

10 

10 

5555 

65 

34 

1845 

225 

62o 

767 

044  602 

134038 

13 

21 

36 

56 

07 

70 

61 

84 

83 

27 

26 

1846 

419 

760 

126 

396  167 

0"  7  201 

01 

32 

32 

20 

99 

17 

16 

08 

08 

'JO 

17 

1847 

613 

9ul 

48  * 

749  643 

681  371 

68 

42 

20 

97 

01 

65 

05 

33 

33 

52 

"9 

1848  B. 

806 

041 

845 

101  118 

454  538 

45 

53 

26 

68 

i»3 

12 

15 

57 

58 

15 

00 

1849 

099 

212 

241 

49ii  619 

23.  t  705 

23 

64 

33 

39 

"0 

63 

71 

86 

86 

77 

02 

1850 

293 

352 

600 

848 

094 

•io3 

871 

00 

75 

20 

00 

10 

in 

20 

10 

10 

40 

31 

1851 

487 

493 

960 

201 

56* 

777 

038 

78 

80 

2rt 

80 

12 

57 

70 

35 

35 

"2 

75 

1852B. 

681 

633 

319 

553044 

r>50  2n5 

55 

'.'6 

•23 

51 

14 

04 

19 

59 

60 

66 

6'i 

1853 

974 

804 

715 

948  545 

326  372 

33 

07 

3n 

22 

20 

55 

70 

8888 

28 

5> 

1854 

168 

944 

074 

300  020 

099  539 

10 

18 

20 

92 

22 

03 

25 

12 

12 

90 

5* 

1855 

361 

085 

434 

653495 

873  705 

87 

2823 

63 

24 

56 

74 

87 

37 

53 

41 

1856  B. 

555 

225 

793 

005  970 

646'872 

65 

3920 

34 

20 

07 

23 

n 

s* 

15 

33 

1857 

848 

396 

189 

400471 

422  ">39 

42 

5027 

05 

32 

48 

80 

00 

90 

78 

24 

1858 

042 

537 

548 

752  947 

195  206 

20 

61  24 

76 

34 

95 

29 

15 

15 

4o 

16 

1859 

236 

677 

908 

105  422 

969372 

07 

71)20 

40 

36 

42 

70 

39  40 

03 

07 

1860  B. 

430 

817 

267 

457 

8y7 

742 

539 

74 

82 

17 

17 

38 

89 

2,8 

6464 

65 

09 

1861 

723 

988 

663 

852 

398 

518 

706 

52 

93 

24 

8N 

44 

41 

84 

92 

02 

28 

01 

1862 

916 

129 

022 

204 

873 

291 

873 

2'.' 

u42060 

46 

vs 

34 

17 

17 

01 

82 

1863 

110 

269 

382 

557 

348 

065 

039 

06  14  17  29 

48 

85 

82 

11 

42 

53 

74 

1864  B. 

304 

409 

741 

909 

823 

83S 

206 

84  25 

14  00 

50 

82 

:-.2 

()-', 

66 

16 

66 

1865 

597 

580 

137 

304 

324 

614 

373 

61 

36 

21 

71 

56 

33 

NO 

05 

i'4 

78 

57 

1866 

791 

721 

496 

657 

799 

387 

540 

39 

47 

17 

42 

58 

80 

88 

19 

10 

il 

40 

1867 

985 

861 

S56 

009 

274 

161 

707 

16 

57 

14 

12 

60 

28 

87 

44 

44 

03 

40 

1868  B. 

178 

001 

215 

362 

749 

934 

873 

U3 

68 

11 

83 

02 

75 

37 

68 

60 

66 

32 

1869 

471 

172 

611 

756 

251 

710 

040 

71 

70 

is 

54 

68 

2i' 

93 

1*7 

97 

•2? 

28 

1870 

665 

313 

970 

109 

726 

483 

207 

48 

On 

15 

20 

60 

73 

43 

21 

21 

01 

15 

1871 

859 

454 

330 

462 

201 

257 

374 

20 

0(» 

12 

97 

71 

20 

93 

46 

46 

53 

o7 

1872  B. 

053 

594 

689 

814 

676 

030 

541 

03 

11 

09 

68 

73 

07 

42 

70 

71 

16 

ON 

1873 

346 

765 

n85 

209 

177 

806 

708 

81 

22 

16 

39 

70 

18 

99 

99 

ytg 

79 

90 

1874 

54o 

905 

484 

561 

652 

579 

875 

5> 

83 

12 

10 

si 

66 

4N 

23 

23 

41 

82 

1875 

733 

046 

804 

914 

127 

353 

41 

35 

43 

09 

80 

96 

13 

97 

48 

48 

04 

73 

1876  B. 

927 

186 

163 

266 

602 

126 

208 

13 

54 

06 

51 

85 

.in 

40 

72 

73 

66 

65 

1877 

220 

357 

559 

661 

103 

902 

375 

90 

05 

13 

•22 

01 

11 

03 

01 

"1 

20 

56 

1878 

414 

498 

918 

013 

579 

675 

542 

6^ 

76 

10 

03 

98 

58 

52 

26 

25 

91 

48 

1879 

608 

638 

278 

366 

54 

449 

708 

45 

8f 

"6 

63 

U5 

05 

02 

50 

51 

54 

30 

1880  B. 

802 

778 

637 

718 

529 

222 

875 

•2^ 

97 

03 

34 

{•- 

52 

53 

75 

75 

16 

31 

1881 

095 

949 

033 

113 

30 

9^8 

042 

On 

08 

10 

05 

03 

04 

t>7 

03 

03 

70 

23 

1882 

288 

90 

392 

465 

505 

771 

209 

77 

19 

06 

77 

05 

51 

57 

28 

28 

42 

14 

1883 

482 

230 

752 

818 

980 

545 

375 

54 

29 

03 

46 

07 

0> 

05 

52 

5:-, 

04 

00 

1884  B. 

676 

370 

111 

170 

455 

318 

542 

32 

40 

10 

17 

00 

45 

55 

77 

77 

67 

07 

1885 

969 

541 

507 

565 

956 

94 

709 

00 

51 

07 

88 

15 

'JO 

12 

06 

05 

20 

89 

TABLE  XXXV. 
Maoris  Epochs. 


Tears. 

Erection. 

Anomaly. 

Variation. 

Longitude. 

1830 

517    412 

11  24  31    4.5 

2  13    2  39 

11  22  55  37.7 

1831 

11    73541 

2  23  14  24.6 

6  22  40    4 

4    21842.8 

1832  B 

428    7  11 

5  21  57  44.4 

11    2  17  28 

8  11  4148.0 

1833 

10  29  57  40 

9    3  44  58.5 

3  24    6  21 

1    4  1528.4 

1834 

4  20  29  1  1 

0    2  28  18.5 

8    3  43  45 

5  13  38  33.6 

1835 

1011    040 

3    1  1138.6 

0  1321  10 

9  23    1  38.8 

1836  B 

4    132    9 

529  5458.7 

4  22  58  34 

2    2  24  44.0 

1837 

10    32239 

911  42  12.8 

9  14  47  27 

6  24  58  24.5 

1838 

3  23  54    9 

0  10  25  32.9 

1  242451 

11    421  29.8 

1839 

9  14  2538 

3    9    853.1 

6    4    2  16 

3  13  44  35.0 

1840  B 

3    4  57    8 

6    7  52  13.2 

10133942 

7  23    7  40.4 

1841 

9    6  47  37 

9  19  39  27.5 

3    52833 

0  1541  20.9 

1842 

2  27  19    7 

0  18  2247.6 

7  15    5  58 

425    426.2 

1843 

8  17  5037 

3  17     6    7.9 

11  24  43  23 

9    4  27  31.6 

1844  B 

2    8  22    7 

6154928.1 

4    4  20  48 

1  135037.0 

1845 

8  10  1236 

9  27  36  42.5 

8  26    9  40 

6    6  24  17.5 

1846 

2    044    6 

0  26  20    2.8 

1    547    5 

10  15  47  23.0 

1847 

7  21  1535 

3  25    3  23.2 

5  15  24  30 

2  25  10  28.3 

1848  B 

1  11  47    5 

6  23  46  43.5 

9  25    1  55 

7    43333.7 

1849 

7  13  37  35 

10    533579 

2  16  50  47 

11  27    7  14.5 

1850 

1494 

1    4  17  18.3 

6  26  28  12 

4    6  30  19.9 

1851 

6  24  40  35 

4    3    038.6 

1165  37 

8  15  53  25.4 

1852  B 

0  15  12    5 

7    1  43  59.2 

3  15  43    3 

0  25  16  31.0 

1858 

6  17    2  34 

10  13  31  137 

8    7  SI  54 

5  17  50  11.6 

1854 

0    7  34    4 

1  12  14*4.1 

0179  20 

9  27  13  17.2 

1855 

5  28    5  33 

4  10  57  54.7 

-4264644 

2    63622.7 

1856  B 

11  18  37    3 

7    9  41  15.2 

9    6  24  10 

6  15  59  28.2 

1857 

5  20  27  33 

10  21  2829.8 

1  28  13    2 

11     833    9.1 

1858 

11  1059    2 

1  20  11  50.3 

6    7  50  27 

3  17  56  14.6 

1859 

5    13033 

4  18  55  10.9 

10  17  27  53 

7  27  1920.1 

1860  B 

10  22    2    3 

7  173831.4 

227    5  18 

0    64225.8 

1861 

4  23  52  32 

10  292546.1 

7  185410 

429  16    6.6 

1862 

10  14  24    2 

1  28    9    6.6 

112831  35 

9    83912.2 

1863 

4    45532 

4265227.S 

4891 

118    217.9 

1864  B 

9  25  27    2 

7  253548.0 

8  174625 

5  27  25  23.5 

1865 

3  27  17  31 

11    723    2.7 

1    93518 

10  19  59    4.3 

1866 

9  17  49    2 

266  23.3 

5  19  1243 

2  29  22  10.1 

1867 

3    8  20  31 

5    44944.0 

9  28  50    9 

7    84515.7 

1868  B 

82852    2 

8    3  33    4.7 

2    82734 

1118    821.4 

1869 

3    0  42  33 

11  15  20  19.6 

7    01626 

4  10  42    2.3 

1870 

8  21  14    2 

2  14    3  40.3 

11    95351 

820    5    8.0 

1871 

2  11  45  33 

5  12  47    0.6 

3  193116 

0  29  28  13.5 

1872  B 

8    217    3 

8  11  3021.2 

7  29    8  42 

5    8  57  19.1 

1873 

2    4    7  32 

11  23  17  35.7 

0  2057  33 

10    1  2459.7 

1874 

7  24  39    2 

322    056.; 

4  20  34  59 

21048    5.3 

1875 

1151031 

5  20  44  16.7 

9  101223 

620  1110.8 

1876  B 

7    542    1 

8  19  27  37.2 

1  19  49  49 

10  29  34  16.3 

1877 

1    73231 

0    11451.8 

6  11  3841 

322    757.2 

1878 

628    4    0 

2  2958  12.3 

10  21  16    6 

8    1  31    2.7 

1879 

0  1835  31 

6  284132.9 

3    05332 

01054    8.2 

1880  B 

6971 

8  27  24  53.4 

7  103057 

1620  17  13.9 

1881 

01057  30 

0    912    8.1 

0    21949 

9125054.7 

1882 

6    1  29    0 

3    7  55  28.6 

4  1157  14 

1  22  14    0.3 

1883 

11  22    0  30 

6    63849.3 

8  21  34  40 

6    1  37    6.8 

1884  B 

5  12  32    0 

9    5  22  10.0 

1    112    4 

1011    011.6 

1885 

11  14  22  29 

0  17    9  24.7 

5  23    0  57 

3    3  33  52.4 

TABLE   XXXV. 
Moon's  H/pochs. 


53 


YSABS. 

Supp.  of  Node. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

s  °  '   " 

s  °  ' 

1830 

6  7  7  11.0 

102446 

498 

502 

900 

904 

427 

062 

025 

433 

1831 

6  26  26  53.3 

215  18 

912 

914 

208 

210 

506 

001 

211 

710 

1832  B 

7  15  4u  35.5 

6  Sou 

326 

327 

5LG 

516 

586 

940 

397 

986 

1833 

859  28.4 

10  7  31 

774 

77'J 

852 

856 

702 

885 

624 

297 

1834 

8  2429  10.7 

128  3 

187 

191 

159 

163 

782 

825 

*10 

573 

1335 

9  13  48  53.0 

51835 

t>01 

6«»3 

407 

469 

861 

764 

996 

850 

1836  B 

10  3  835.2 

998 

015 

016 

775 

775 

941 

703 

182 

127 

1837 

lu  22  31  28.1 

1  1049 

463 

468 

111 

116 

057 

648 

409 

437 

1838 

11  11  51  10.4 

5  121 

876 

880 

419 

423 

137 

£88 

5y5 

714 

1839 

0  1  10  52.6 

821  53 

290 

292 

726 

729 

217 

527 

781 

991 

1840  B 

0  20  30  34.9 

01225 

704 

705 

034 

035 

296 

466 

967 

268 

1841 

1  9  53  27.7 

414  6 

152 

157 

370 

376 

412 

411 

194 

578 

1842 

I  29  13  10.0 

8  438 

566 

569 

678 

682 

492 

360 

360 

855 

1843 

2  18  32  52.2 

11  25  lo 

980 

980 

986 

988 

572 

290 

566 

131 

1844  B 

3  7  52  34.5 

3  1542 

393 

394 

293 

294 

651 

229 

752 

408 

1845 

3  27  lp  27.4 

71723 

840 

846 

629 

634 

767 

174 

979 

718 

1846 

4  16  35  9.6 

11  755 

254 

258 

937 

941 

847 

113 

165 

995 

1847 

5  55451.8 

2  28  27 

668 

670 

245 

247 

927 

053 

351 

272 

1848  B 

5  25  1434.1 

6  1859 

082 

083 

553 

553 

006 

992 

537 

549 

1849. 

6  1437  27.0 

10  20  40 

531 

535 

889 

893 

122 

937 

764 

859 

1850 

7  357  9.2 

2H12 

944 

947 

196 

200 

202 

876 

950 

136 

1851 

7  23  16  51.5 

6  144 

358 

359 

504 

506 

282 

816 

136 

413 

1852  B 

8  123633.6 

92217 

772 

772 

812 

812 

362 

755 

322 

689 

1853 

9  1  59  26.5 

12358 

220 

2J3 

148 

152 

477 

700 

549 

000 

1854 

9  21  19  8.8 

5  U30 

634 

636 

456 

459 

557 

639 

735 

276 

1855 

10103851.1 

952 

047 

048 

763 

765 

637 

579 

921 

553 

1856  B 

10  29  58  33.3 

02534 

461 

461 

071 

071 

717 

518 

107 

830 

1857 

11  192126.2 

42715 

909 

912 

407 

411 

832 

463 

334 

140 

1858 

0  841  8.4 

81747 

323 

325 

715 

718 

912 

402 

520 

417 

1859 

028  050.7 

0  8  19 

736 

737 

023 

024 

992 

342 

706 

694 

1860  B 

1  17  2032.9 

32851 

150 

150 

330 

330 

072 

281 

892 

971 

1861 

2  64325.8 

8  032 

598 

601 

666 

670 

187 

226 

119 

281 

1862 

2  26  3  8.0 

1121  4 

012 

014 

974 

977 

267 

165 

305 

558 

1863 

3  152250.1 

3H36 

426 

426 

282 

283 

347 

105 

491 

834 

1864  B 

4  44232.3 

728 

839 

839 

590 

589 

427 

044 

677 

111 

1865 

424  525.2 

11  349 

287 

291 

926 

929 

542 

989 

904 

422 

1866 

51325  7.3 

22421 

701 

703 

233 

236 

622 

928 

090 

698 

1867 

6  24449.5 

61453 

115 

115 

541 

542 

702 

868 

276 

975 

1868  B 

622  431.7 

10  526 

529 

528 

849 

848 

782 

807 

462 

252 

1869 

7  11  27  24.6 

277 

977 

980 

185 

188 

897 

752 

689 

562 

1870 

8  047  6.7 

527  39 

390 

392 

493 

495 

977 

691 

875 

839 

1871 

820  649.0 

91811 

804 

804 

801 

801 

057 

631 

061 

116 

1872  B 

9  92631.1 

1  844 

218 

217 

109 

107 

137 

570 

247 

392 

1873 

9  28  49  24.0 

5  1025 

666 

668 

445 

447 

252 

515 

474 

703 

1874 

10  18  9  6.3 

9  057 

080 

081 

753 

754 

332 

454 

660 

979 

1875 

11  72848.6 

02129 

493 

493 

060 

060 

412 

394 

846 

256 

1876  B 

11  26  48  30.8 

412  1 

907 

906 

368 

366 

492 

333 

032 

533 

1877 

016  11  23.7 

9  13  42 

355 

357 

704 

706 

607 

278 

259 

843 

1878 

1  5  31  5.9 

0  414 

769 

770 

012 

013 

687 

217 

445 

120 

1879 

1  24  50  48.2 

32446 

182 

182 

320 

319 

767 

157 

631 

397 

1880  B 

2  14  lu  80.4 

7  15  18 

596 

595 

627 

623 

847 

096 

817 

674 

1881 

3  3  83  23.3 

11  1659 

044 

046 

963 

965 

962 

041 

044 

984 

1882 

32253  5.5 

3  7  31 

458 

459 

271 

272 

042 

980 

230 

261 

1883 

412  1247.6 

628  3 

872 

871 

579 

578 

122 

920 

416 

537 

1884  B 

5  1  32  29.8 

101835 

2»5 

284 

887 

884 

202 

859 

602 

814 

1885 

5  20  55  23.0 

22016 

733 

736 

223 

224 

317 

804 

829 

125 

54 


TABLE  XXXVI 
Moon's     Motions  for  Months. 


Months. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

January 

00000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000  000 

OOOiOOO 

000 

February 

08487 

0146 

2246 

8896 

0402 

1533 

1789 

2099 

0753 

175 

965  184 

059 

,,     ,    (  Com. 

16153 

8343 

1371 

6931 

9797 

1951 

3404 

3027 

1433 

139 

836 

157 

016 

March  {  Bis. 

16427 

8993 

2411 

7218 

0132 

2323 

3462 

3418 

1457 

209 

868 

228 

050 

•i    5  Com- 

24640 

8490 

3616 

5827 

0199 

3484 

5193 

5126 

2186 

314 

801 

342 

076 

•    P        \  Bis 

24914 

9140  4657 

6114 

0534 

3856 

5251 

5517 

2210 

384 

832 

412 

110 

,,          (  Com. 

32853 

7986  4822 

4436 

026514646  6924 

6835 

2914 

419 

735 

456 

101 

May  Us. 

33127 

8636  5862  4723 

0600 

5018  6982 

7226 

2938 

489 

766 

526 

135 

T            (  Com. 

41340 

8133 

7067 

3332 

0666 

6179 

8713 

8934'3667 

593 

700 

640 

160 

June     <  g  g 

41614 

8783 

8107 

3619 

1002 

6551  8771 

9325  3691 

663 

731 

710 

194 

T  ,        (  Com. 

July  Us. 

49554 
49828 

7629 

8279 

8273 
9313 

1942 

2228 

0732 

1068 

7341 
7713 

0444 
0502 

0643  4396 
1034  4420 

698 
768 

634 
665 

754 

824 

185 
219 

.           (  Com. 

58041 

7776 

0518 

0838 

113488742233 

2742 

5148 

873 

599 

938 

245 

Aus-   IBIS. 

58315 

8426 

1558 

1125 

14709246  2290 

31335173 

943 

630 

009 

279 

c           (  Com. 

66528 

7922 

2764 

9734 

1536  0408  4021 

4842  5901 

048 

563 

123 

304 

S€T*    {Bis. 

66802 

8572 

3804 

0021 

1871 

0780 

4079 

5232 

5925  118 

595 

193 

338 

xv  .        <  Com. 

74741 

7419 

3969 

8343 

1602 

1569 

5752 

6550  6630  152 

497 

237 

329 

Oct.      ^  Bi<, 

75015 

8069 

5009 

8630 

1938 

1941 

5810 

6941 

6654!  222 

528 

307 

363 

NT          (  Com. 

83228 

7565 

6215 

7239 

2004 

3102 

7541 

8649 

7382  327 

462 

421 

383 

Nov'     {Bis. 

8350,; 

8215 

7255 

7526  2339 

3475 

7599 

9040 

74071397 

493 

492 

423 

Dec       (  Com. 

91442 

7062 

7420 

584812070 

4264 

9272 

0358 

8111  432 

396 

535 

414 

91716 

7712 

8460 

6135|2405 

4636 

9330 

0749 

8135  502 

427 

606 

448 

TABLE  XXXVI. 

Moon's 

Motions 

for  Months. 

Months. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

January 

8       0         '        " 

0000 

000 

0.0 

000 

0 

0    0    0    0.0 

February 

11  20  48  42 

1   15     0  53.1 

0  17  54 

48 

1  18  28     5.8 

March  J  Com' 

10     7  40  26 

1  20  50 

4.2 

11  29  15 

15 

1  27  24  26.6 

(  B:s. 

10  18  59  26 

2     3  53  58.2 

0  11  26 

42 

2  10  35     1.6 

A     -i     j  Com. 

9  28  29     8 

3     5  50  57.3 

0  17  10 

3 

3  15  52  32.5 

"        {  Bis. 

10    9  48     8 

3  18  54  51.2 

0  29  21 

29 

3  29     3     7.5 

U         J  Com- 

9     7  58  51 

4     7  47  56.4 

0  22  53 

24 

4  21   10     3.3 

'       (  Bis. 

9  19  17  50 

4  20  51  50.3 

5     4 

50 

5     4  20  38  3 

,            i  Com. 

8  28  47  33 

5  22  48  49.4 

10  48 

11 

6     9  38     9.1 

June     }Bis. 

9  10    6  33 

6     5  52  43.4 

22  59 

38 

6  22  48  44.1 

t  ,        <  Com. 

8    8  17  16 

6  24  45  48.5 

16  31 

32 

7  14  55  39.9 

July  iBis. 

8  19  36  15 

7     7  49  42.5 

28  42 

59 

7  28     6  15.0 

A          (  Com. 

7  29     5  59 

8     9  46  41.6 

2     4  26 

20 

9     3  23  45.8 

8  10  24  58 

8  22  50  35.5 

2  16  37 

47 

9  16  34  20.8 

««,«!      5  Com. 

7  19  54  41 

9  24  47  34.6 

2  22  21 

7 

10  21  51  51.6 

oept.     <  TJ. 
r        (  His. 

8     1   13  40 

10     7  51  28.6 

3    4  32 

34 

11     5     2  26.7 

Oct       J  Com- 

6  29  24  24 

10  26  44  33.7 

2  28     4 

28 

11  27     9  22.4 

(  Bis. 

7  10  43  23 

11     9  48  27.7 

3  10  15 

55 

0  10  19  57.5 

Nov      i  Corn' 

6  20  13     6 

0  11  45  26.8 

3  15  59 

16 

1   15  37  28.3 

(  bis. 

7     1  32     5 

0  24  49  20.7 

3  28  10 

43 

1  28  48     3.3 

Dec      i  Coin' 

5  29  42  49 

1  13  42  25.9 

3  21  42 

37 

2  20  54  59.1 

\  Bis. 

6  11   1     48 

1  26  46  19.8 

4     3  54 

4 

345  34.1 

TABLE  XXXVI. 
Moon's     Motions  for  Months. 


55 


Months. 

14 

15 

16 

17 

18    ] 

9 

;  2 

0 

21122 

23124,25126  27 

28 

2930 

31 

January 

000 

000 

000  000  000  000 


000 

oo'oo 

oo  oo  ooloo  oo 

00 

00 

00 

__ 
00 

February 

074(946 

135  304 

805  066 

014 

:24.'26|14J8228|14  17 

2!J 

96 

05 

07 

M      ,    c  Com. 

851 

801 

159 

482  532  125 

027 

4550  985743  18:  12  46 

82 

10 

15 

rch  \  Bis. 

950 

831 

196  524 

558,127 

027 

46  51 

0*694*21 

19 

51 

8fi 

10 

15 

A  ^     (  Com. 

925 

747 

294  786 

336!191 

041 

Un 

1239  70  '32 

29 

76 

7? 

1523 

024 

778  331 

828 

362  193 

042 

6977 

22  42  174  136  13680 

90 

1623 

M         5  Com- 

899 

663 

392  047 

115254055 

91  02  15  199443  38  01 

?o 

21 

30 

May      1  Bis. 

999 

693  429  089 

141  256 

055 

9203 

26  22  98  47  45  05 

73 

21 

30 

T           j  Com. 

973 

609  527  351 

920  320  069  15^8 

29  01  21  57 

5531 

65 

2638 

June     <  -p 

073 

639 

563 

393 

946  322069 

15>29 

400425.61 

62  35 

6« 

2638 

July      (  Com. 

948 

525 

625 

613 

699^84083 

3754 

3381  ' 

is'es 

6456 

58 

31  45 

047 

555 

661 

655 

725  386 

083 

3855 

4384|4972 

71 

60 

81 

31 

46 

(  Com. 

022 

471 

759 

917 

503  449 

097 

61  80 

47164' 

r282 

81 

85 

53 

36 

53 

£'     \  Bis. 

121 

501 

796 

959 

529  451 

097 

62  81 

5766' 

r?86 

83 

90 

M 

36 

53 

c     .      (  Com. 

096 

417 

894 

221 

308  515 

111 

8507 

61J46( 

)0  97 

97 

15 

49 

4261 

SePl-    }  Bis. 

195 

447 

931 

263 

334517 

111 

85 

0* 

7149( 

)401 

04 

1!) 

4261 

0          (  Com. 

071 

333 

992 

483 

087  578 

125 

07 

32 

65265 

S3  08 

07 

40 

4! 

4768 

Uct>      1  Bis. 

170 

363 

029 

525 

113581 

126 

08 

32 

11 

.   OQ  < 

•vO    * 

IS  11 

1444 

41 

4769 

Nov      i  Com' 

145 

279 

127 

787 

892644 

139 

31 

59 

79  08(51  22 

23 

70 

i7 

52 

76 

^ov'     \  Bis. 

244 

309 

163 

829 

918  646 

140 

32 

60 

89  IK 

5526 

30 

74 

10 

52 

76 

Dec      J  Com' 

120 

194225 

049  670  708 

153 

54 

85 

8388' 

r433 

3395 

29 

5784 

1  Bis. 

219 

225*261 

091 

696  710 

153 

54 

86 

93  90  79  37 

40  99 

32 

5784 

TABLE  XXXVI. 

Moon's  Motions  for 

Months. 

Months. 

Supp.  of  Node.l 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI  XII 

»       0 

/ 

s 

0          ' 

January 

0    0 

0     0.0 

0 

0     0 

000 

000 

000 

000 

000 

000 

000  000 

February 

0     1  38  29.7 

11  15  43 

054 

224 

875 

045 

111 

165 

290   043 

March  $  Com* 

0     3 

7  27.5 

9  27  59 

;  007  330 

666 

989 

114 

313 

455   984 

{  Bis. 

0     3  10  38.2 

10 

9     8 

,041 

369 

694 

023 

150   319 

496   018 

April     |Com- 

0     4  45  57.3 

9  13  42 

061 

554 

542 

034 

225 

478 

745   027 

0    4  49    7.9 

9  24  51 

095 

593 

570 

068 

261  i  484 

787   061 

M          5  Com- 

0     6  21  16.4 

8  18  15 

081 

738 

389 

046 

300   638 

993   036 

Yiay      <  Tj  • 
(  Jois. 

0     6  24  27.0 

8  29  25 

115 

778 

417 

080 

336   643 

034   070 

June     $  Com- 

0     7  59  46.1 

8 

3  58 

136 

962 

264 

091 

411 

802 

282  !  079 

(  Bis. 

0    8 

2  56.7 

8  15    8 

170 

002 

293 

124 

447 

808 

324 

113 

Tnlv        J  ^°m 

0     9  5 

5     5.2      7 

8  32 

156 

147 

112 

103 

486 

962 

531 

088 

Juiy      \  Bis. 

0     9  38  15.9      7  19  41 

190 

186 

140 

136 

522 

967 

572  !  122 

A          (  Com. 

0  11   13  35.0 

6  24  15 

210 

371 

987 

147 

597 

126 

820    131 

u§>'      (  Bis. 

0  11  16  45.6 

7 

5  24 

244 

411 

015 

182 

633    132 

862    164 

o*5  Com. 

0  12  52    4.7 

6 

9  58 

265 

595 

862 

193 

708  i  291 

110    173 

ep       <  Bis 

0  12  55  15.4 

6  21     7 

299 

635 

891 

227 

744 

296 

152   207 

n  .        (  Com. 

0  14  27  23.8 

5  14  32 

285 

780 

710 

204 

783   451 

358 

182 

Oct"      i  Bis. 

0  14  30  34.4 

5  25  41 

319 

819 

738 

238  1  819   456 

400   216 

Nnv      J  Com. 

0  16 

5  53  5 

5 

0  15 

339 

004 

585 

250  1  894  i  615    648    225 

*>UV.          S    -r> 

0  16 

9     4.2 

5  11  24 

373 

043 

613 

283 

930   621(690 

259 

-P.           (  Com. 

0  17  41   12.6 

4 

4  49 

359 

188 

432 

261 

969    775    896 

234 

Dec.     <  g-g 

0  17  44  23.3 

4  15  58 

393 

228 

461 

295 

005  1  780  ||  938 

268' 

56 


TABLE  XXXVII. 


Moon's  Motions  for  Days. 


D. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

1 

00000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

000 

000 

000 

000  i 

2 

00274 

0650 

1040 

0287 

0336 

0372 

0058 

0390 

0024 

070 

031 

070 

034 

3 

00548 

1300 

2080 

0574 

0671 

0744 

0115 

0781 

0049 

140 

062 

141 

068  ! 

4 

00821 

1950 

3121 

0861 

1007 

1116 

0173 

1171 

0073 

210 

093 

211 

103 

5 

01095 

2600 

4161 

1148 

1342 

1488 

0231 

1561 

0097 

281 

125 

282 

137 

6 

01369 

3249 

5201 

1435 

1678 

1860 

0289 

1952 

0121 

351 

156 

352 

171 

7 

01643 

3899 

6241 

1722 

2013 

2232 

0346 

2342 

0146 

421 

187 

423 

205 

8 

01916 

4549 

7281 

2009 

2349 

2604 

0404 

2732 

0170 

491 

218 

493 

239 

9 

02190 

5199 

8321 

2296 

2684 

2976 

0462 

3122 

0194 

561 

249 

564 

273 

10 

02464 

5849 

9362 

2583 

3020 

3348 

0519 

3513 

0219 

631 

280 

634 

308 

11 

02738 

6499 

0402 

2870 

3355 

3720 

0577 

3903 

0243 

702 

311 

705 

342 

12 

03012 

7149 

1442 

3157 

3691 

4093 

0635 

4293 

0267 

772 

342 

775 

376 

13 

03285 

7799 

2482 

3444 

4026 

4465 

0692 

4684 

0291 

842 

374 

845 

410 

14 

03559 

8449 

3522 

3731 

4362 

4837 

0750 

5074 

0316  '912 

405 

916 

444 

15 

03833 

9098 

4563 

4018 

4698 

5209 

0808 

5464 

0340  !  982 

436 

986 

478 

1G 

04107 

9748 

5603 

4305 

5033 

5581 

0866 

5854 

0364  '  052 

467 

057 

513 

17 

04380 

0398 

6643 

4592 

5369 

5953 

0923 

6245 

0389  1  122 

498 

127 

547 

18 

04654 

1048 

7683 

4878 

5704 

6325 

0981 

6635 

0413 

193 

529 

198 

581 

19 

04928 

1698 

8723 

5165 

6040 

6697 

1039 

7025 

0437 

263 

560 

268 

315 

20 

05202 

2348 

9763 

5452 

6375 

7069 

1096 

7416 

0461 

333 

591 

339 

649 

21 

05476 

2998 

0804 

5739 

6711 

7441 

1154 

7806 

0486 

403 

623 

409 

683 

22 

05749 

3648 

1844 

6026 

7046 

7813 

1212 

8196 

0510 

473 

654 

480 

718 

23 

06023 

4298 

2884 

6313 

7382 

8185 

1269 

8586 

0534 

543 

685 

550 

752 

24 

06297 

4947 

3924 

6600 

7717 

8557 

1327 

8977 

0559 

614 

716 

621 

786 

25 

06571 

5597 

4964 

6887 

8053 

8929 

1385 

9367 

0583 

684 

747 

691 

820 

26 

06844 

6247 

6005 

7174 

8389 

9301 

1443 

9757 

0607 

754 

778 

762 

854 

27 

07118 

6897 

7045 

7461 

8724 

9673 

1500 

0148 

0631 

824 

809 

832 

888 

28 

07392 

7547 

8085 

7748 

9060 

0045 

1558 

0538 

0656 

894 

840 

903 

923 

29 

07666 

8197 

9125 

8035 

9395 

0417 

1616 

0928 

0680 

964 

872 

973 

957 

30 

07940 

8847 

0165 

8322 

9731 

0789 

1673 

11-19 

0704 

034 

903 

043 

991 

3'. 

08213 

9497 

1205 

8609 

0066 

1161 

1731 

1709 

0729 

105 

934 

114 

025 

TABLE  XXXVII 


57 


Moon's  Motion  for  Days. 


D 

14 

15 

16 

17 

18 

19 

20 

21 

22  23 

24 

25 

26 

27 

28 

29 

30 

31 

1 

ooo  "oob 

000 

000 

000 

000 

000 

« 

00 

00 

00 

00 

00 

00 

00 

00 

00 

00 

o 

099  031 

037 

042 

026 

002 

000 

!°1 

01 

10 

03 

04 

04 

07 

04 

03 

00 

00 

3 

198 

061 

073 

084 

052 

004 

001 

;o2 

02 

20 

05 

08 

07 

14 

08 

06 

00 

00 

4 

297 

092 

110 

126 

078 

006 

00] 

02 

03 

30 

08 

12 

11 

21 

13 

09 

01 

01 

5 

397 

122 

146 

168 

104 

008 

002 

03 

03 

41 

11 

16 

15 

28 

17 

12 

01 

01 

496 

153 

183 

210 

130 

Oil 

002 

04  04 

51 

13 

21 

18 

35 

21 

15 

01 

01 

7 

595  |  183 

220  1  252 

156 

013 

003 

;05 

05 

61 

16 

25 

22 

42 

25 

18 

01 

01 

8 

694  i  214 

256  294 

182 

015 

003 

05 

06 

71 

19 

29 

26 

49 

29 

22 

01 

02 

9 

793  1  244 

293 

336 

208 

017 

004 

06 

07 

81 

21 

33 

30 

56 

33 

25 

01 

02 

10 

892 

275 

329 

379 

234 

019 

004 

07 

08 

91 

24 

37 

33 

63 

38 

28 

02 

02 

11 

992 

305 

366 

421 

260 

021 

005 

08 

09 

01 

27 

41 

37 

70 

42 

31 

02 

02 

12 

091 

336 

403 

463 

286 

023 

005 

08 

09 

11 

29 

45 

41 

77 

46 

34 

02 

03 

13 

190 

366 

439 

505 

312 

025 

005 

09 

10 

22 

32 

49 

44 

84 

50 

37 

02 

03 

14 

289 

397 

476 

547 

337 

028 

006 

10 

11 

32 

34 

53 

48 

91 

54 

40 

02 

03 

15 

388 

427 

512 

589 

363 

030 

006 

11 

12 

42 

37 

58 

52 

98 

58 

43 

02 

03 

16  487 

458 

549 

631 

389 

032 

007 

11 

13 

52 

40 

62 

55 

05 

63 

46 

03 

04 

17|  587 

488 

586  673 

415 

034 

007 

12 

14 

62 

42 

66 

59 

12 

67 

49 

03 

04 

18  686 

519 

622 

715  441 

036 

008 

13 

14 

72 

45 

70 

63 

19 

71 

52 

03 

04 

19 

785 

549 

659 

757  467 

038 

008 

14 

15 

82 

48 

74 

66 

26 

75 

55 

03 

04 

20 

884 

580 

695 

799  493 

040  009 

14 

16 

92 

50 

78 

70 

33 

79 

59 

03 

05 

21 

983 

611 

732 

841  519 

042  009 

15 

17 

03 

53 

82' 

74 

40 

84 

62 

03 

05 

22 

082 

641 

769  883  545 

044  010 

16 

18 

13 

56 

86 

77 

47 

88 

65 

04 

05 

23 

182 

672 

8051925  571 

047,010 

17 

19 

23 

58 

90 

81 

54 

92 

68 

04 

05 

24 

281  702 

842 

967  597 

049:011 

17 

20133 

61 

95 

85 

61 

96 

71 

04 

06 

25 

380 

733 

878 

009  623 

051  1  Oil 

18 

20 

43 

64 

99 

89 

68 

00 

74 

04 

06 

26  !  479 

763 

915  052  649 

053  'Oil 

19 

21 

53 

66 

03 

92 

75 

04 

77 

04 

06 

27  578  j  794  952  094  675 

055  012 

20 

22 

63 

69 

07 

96 

82 

09 

SO 

04 

06 

28:677  824  !  988  j  136  701 

057 

012 

20 

23 

73 

72 

11 

00 

89 

13 

S3 

05 

06 

29  j  777  1855  0251178  727 

059 

0131 

21 

24 

84 

74 

15 

03 

96 

17 

86 

05 

07 

30  876  8851061 

220  753 

061 

013 

22 

25 

94 

77 

19 

07 

03 

21 

89 

05 

07 

31  975  1  916  |  098 

262  779 

064  014 

23 

26 

04 

80 

23 

11  10 

25 

92 

05 

07 

58 


TABLE  XXXVII. 


Moon's  Motions  for  Days. 


D. 

Evection. 

Anomaly. 

Variation. 

M.  Longitude. 

a     o      ' 

s      o      /     " 

a    o     '    " 

»';>*'" 

1 

0000 

0    0    0    00 

0000 

0    0    0    00 

2 

0  11   18  59 

0  13    3  54.0 

0  12  11  27 

0  13  10  35.0 

3 

0  22  37  59 

0  26     7  47.9 

0  24  22  53 

0  26  21  10.1 

4 

1     3  56  58 

1     9  11  41.9 

1     6  34  20 

1     9  31  45.1 

5 

1  15  15  58 

1  22  15  35.9 

1  18  45  47 

1  22  42  20.1 

6 

1  26  34  57 

2     5  19  29.8 

2    0  57  13 

2     5  52  55.1 

7 

2     7  53  57 

2  18  23  23.8 

2  13    8  40 

2  19     3  30.2 

8 

2  19  12  56 

3     1  27  17.8 

2  25  20     7 

3    2  14    5.2 

9 

3    0  31  55 

3  14  31  11.7 

3    7  31  34 

3  15  24  40.2 

10 

3  11  50  55 

3  27  35    5.7 

3  19  43    0 

3  28  35  15.2 

11 

3  23    9  54 

4  10  38  59.7 

4     1  54  27 

4  11  45  50.3 

12 

4    4  28  54 

4  23  42  53.7 

4  14    5  54 

4  24  56  25.3 

13 

4  15  47  53 

5     6  46  47.6 

4  26  17  20 

5    8    7    0.3 

14 

4  27    6  53 

5  19  50  41.6 

5    8  28  47 

5  21  17  35.4 

15 

5    8  25  52 

6    2  54  35.6 

5  20  40  14 

6    4  28  10.1 

16 

5  19  44  51 

6  15  58  29.5 

6    2  51  40 

6  17  38  45.4 

17 

6     1    3  51 

6  29    2  23.5 

6  15    3    7 

7    0  49  20.4 

18 

6  12  22  50 

7  12    6  17.5 

6  27  14  34 

7  13  59  55.5 

19 

6  23  41  50 

7  25  10  11.4 

7    9  26     1 

7  27  10  30.5 

20 

7    5    0  49 

8    8  14    5.4 

7  21  37  27 

8  10  21     5.5 

21 

7  16  19  49 

8  21  17  59.4 

8    3  48  54 

8  23  31  40.5 

22 

7  27  38  48 

9    4  21  53.4 

8  16    0  21 

9     6  42  15.6 

23 

8    8  57  47 

9  17  25  47.3 

8  28  11  47 

9  19  52  50.6 

24 

8  20  16  47 

10     0  29  41.3 

9  10  23  14 

10    3    3  25.6 

25 

9     1  35  46 

10  13  33  35.3 

9  22  34  41 

10  16  14    0.7 

26 

9  12  54  46 

10  26  37  29.2 

10    4  46    7 

10  29  24  35.7 

27 

9  24  13  45 

11     9  41  23.2 

10  16  57  34 

11  12  35  10.7 

28 

10    5  32  45 

11  22  45  17.2 

10  29    9     1 

11  25  45  45.7 

29 

10  16  51  44 

0    5  49  11.1 

11  11  20  28 

0    8  56  20.8 

30 

10  28  10  43 

0  18  53    5.1 

11  23  31  54 

0  22    6  55.8 

31 

11    9  29  43 

1     1  56  59.1 

0     5  43  21 

1     5  17  30.8 

TABLE.   XXXVII. 


59 


Moon's  Motions  for  Days. 


D 

Supp.  of  Node. 

II 

v  • 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

s  o   '  " 

«  °  ' 

I 

00  0  0.0 

000 

000 

000 

000 

000 

000 

000 

000 

000 

2 

003  10.6 

0  11  9 

034 

039 

028 

034 

036 

005 

042 

034 

3 

006  21.3 

0  22  18 

068 

079 

056 

067 

072 

Oil 

083 

067 

4 

009  31.9 

1  3  27 

102 

118 

085 

101 

108 

016 

125 

101 

5 

0  0  12  42.5 

1  14  37 

136 

158 

113 

135 

143 

021 

166 

135 

6 

0  0  15  53.2 

1  25  46 

170 

197 

141 

169 

179 

027 

208 

168 

7 

0  0  19  3.8 

2  6  55 

204 

237 

169 

202 

215 

032 

250 

202 

S 

0  0  22  14.5 

2  18  4 

238 

276 

198 

236 

251 

037 

291 

235 

9 

0  0  25  25.1 

2  29  13 

272 

316 

226 

270 

287 

043 

333 

269 

10 

0  0  28  35.7 

3  10  22 

306 

355 

254 

303 

323 

048 

374 

303 

11 

0  0  31  46.4 

3  21  31 

340 

395 

282 

337 

358 

053 

416 

336 

12 

0  0  34  57.0 

4  2  40 

374 

434 

311 

371 

394 

058 

458 

370 

13 

0  0  38  7.6 

4  13  50 

408 

474 

339 

405 

430 

064 

499 

404 

14 

0  0  41  18.3 

4  24  59 

442 

513 

367 

438 

466 

069 

541 

437 

15 

0  0  44  28.9 

568 

476 

553 

395 

472 

502 

074 

583 

471 

16 

0  0  47  39.5 

5  17  17 

510 

592 

424 

506 

538 

080 

624 

505 

17 

0  0  50  50.2 

5  28  26 

544 

632 

452 

539 

573 

085 

666 

538 

IS 

0  0  54  0.8 

6  9  35 

578 

671 

480 

573 

609 

090 

707 

572 

19 

0  0  57  11.5 

6  20  44 

612 

711 

508 

607 

645 

096 

749 

605 

20 

0  1  0  22.1 

7  1  53 

646 

750 

537 

641 

681 

101 

791 

639 

•21 

0    3  32.7 

7  13  3 

680 

790 

565 

674 

717 

106 

832 

673 

22 

0    6  43.4 

7  24  12 

714 

829 

593 

708 

753 

112 

874 

706 

23 

0    9  54.0 

8  5  21 

748 

869 

621 

742 

788 

117 

915 

740 

24 

0   13  4.6 

8  16  30 

782 

908 

650 

775 

824 

122 

957 

774 

25 

0   16  15.3 

8  27  39 

816 

948 

678 

809 

860 

128 

999 

807 

20 

0   19  25.9 

9  8  48 

850 

987 

706 

843 

896 

133 

040 

841 

27 

0   22  36.5 

9  19  57 

884 

027 

734 

877 

932 

138 

082 

875 

28 

0   25  47.2 

10  1  6 

918 

066 

762 

910 

968 

143 

123 

908 

29 

0   28  57.8 

10  12  16 

952 

106 

791 

944 

003 

149 

165 

942 

30 

0   32  8.5 

10  23  25 

986 

145 

819 

978 

039 

154 

207 

975 

.3l 

0   35  19.1 

11  4  34 

020 

185 

847 

Oil 

075 

159 

248 

009 

60 


TABLE  XXXVIII. 

Moon's  Motions  for  Hours. 


H. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

1 

11 

27 

43 

12 

14 

16 

2 

16 

1 

3 

1 

3 

1 

2 

23 

54 

87 

24 

28 

31 

5 

33 

2 

6 

3 

6 

3 

3 

34 

81 

130 

36 

42 

47 

7 

49 

3 

9 

4 

9 

4 

4 

46 

108 

173 

48 

56 

62 

10 

65 

4 

12 

5 

12 

6 

5 

57 

135 

217 

60 

70 

78 

12 

81 

5 

15 

6 

15 

7 

6 

68 

162 

260 

72 

84 

93 

14 

98 

6 

18 

8 

18 

9 

7 

80 

190 

303 

84 

98 

109 

17 

114 

7 

20 

9 

20 

10 

8 

91 

217 

347 

96 

112 

124 

19 

130 

8 

23 

10 

23 

11 

9 

103 

244 

390 

108 

126 

140 

22 

146 

9 

26 

12 

26 

13 

10 

114 

271 

433 

120 

140 

155 

24 

163 

10 

29 

13 

29 

14 

11 

125 

298 

477 

131 

154 

171 

26 

179 

11 

32 

14 

32 

16 

12 

137 

325 

520 

143 

168 

186 

29 

195 

12 

35 

16 

35 

17 

13 

148 

352 

563 

155 

182 

202 

31 

211 

13 

38 

17 

38 

18 

14 

160 

379 

607 

167 

196 

217 

34 

228 

14 

41 

18 

41 

20 

15 

171 

406 

650 

179 

210 

233 

36 

244 

15 

44 

19 

44 

21 

16 

182 

433 

693 

191 

224 

248 

38 

260 

16 

47 

21 

47 

23 

17 

194 

460 

737 

203 

238 

264 

41 

276 

17 

50 

22 

50 

24 

18 

205 

487 

780 

215 

252 

279 

43 

293 

18 

53 

23 

53 

25 

19 

217 

515 

823 

227 

266 

295 

46 

309 

19 

56 

25 

56 

27 

20 

228 

542 

867 

239 

280 

310 

48 

325 

20 

58 

26 

58 

28 

21 

239 

569 

910 

251 

294 

326 

50 

341 

21 

61 

27 

61 

30 

22 

251 

596 

953 

263 

308 

341 

53 

358 

22 

64 

28 

64 

31 

23 

262 

623 

997 

275 

322 

357 

55 

374 

23 

67 

30 

67 

33 

24 

274 

650 

1040 

287 

336 

372 

58 

390 

24 

70 

31 

70 

34 

Hours. 

Evection. 

Anomaly. 

Variation. 

Longitude. 

o  /  » 

O    '    '/ 

0    '    " 

0    '    " 

1 

0  28  17 

0  32  39.7 

0  30  29 

0  32  56.5 

2 

0  56  35 

1  5  19.5 

1  0  57 

1  5  52.9 

3 

1  24  52 

1  37  59.2 

1  31  26 

1  38  49.4 

4 

1  53  10 

2  10  39.0 

2  1  54 

2  11  45.8 

5 

2  21  27 

2  43  18.7 

2  32  23 

2  44  42.3 

6 

2  49  45 

3  15  58.5 

3  2  52 

3  17  38.8 

7 

3  18  2 

3  48  38.2 

3  33  20 

3  50  35.2 

8 

3  46  20 

4  21  18.0 

4  3  49 

4  23  31.7 

9 

4  14  37 

4  53  57.7 

4  34  17 

4  56  28.1 

10 

4  42  55 

5  26  37.5 

5  4  46 

5  29  24.6 

11 

5  11  12 

5  59  17.2 

5  35  15 

6  2  21.0 

12 

5  39  30 

6  31  57.0 

6  5  43 

6  35  17.5 

13 

6  7  47 

7  4  36.7 

6  36  12 

7  8  14.0 

14 

6  36  5 

7  37  16.5 

7  6  40 

7  41  10.4 

15 

7  4  22 

8  9  56.2 

7  37  9 

8  14  6.9 

16 

7  32  40 

8  42  36.0 

8  7  38 

8  47  3.4 

17 

8  0  57 

9  15  15.7 

8  38  6 

9  19  59.8 

18 

8  29  15 

9  47  55.5 

9  8  35 

9  52  56.3 

19 

8  57  32 

10  20  35.2 

9  39  3 

10  25  52.7 

20 

9  25  50 

10  53  15.0 

10  9  32 

10  58  49.2 

21 

9  54  7 

11  25  54.7 

10  40  1 

11  31  45.6 

22 

10  22  24 

11  58  34.5 

11  10  29 

12  4  42.1 

23 

10  50  42 

12  31  14.2 

11  40  58 

12  37  38.6 

24 

11  18  59 

13  3  54.0 

12  11  27 

13  10  35.0 

TABLE.   XXXVIII. 


61 


Moorfs  Motions  for  Hours. 


H. 

14 

15 

16 

17 

18  ;  19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

1 

4 

1 

2 

2 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

8 

3 

3 

4 

2 

0 

0 

0 

0 

1 

0 

0 

0 

1 

0 

0 

3 

12 

4 

5 

5 

3 

1  ° 

0 

0 

0 

1 

0 

1 

0 

1 

1 

0 

4 

16 

5 

6 

7 

4 

0 

0 

0 

0 

2 

0 

1 

1 

1 

1 

1 

5 

21 

6 

8 

9 

5 

0 

0 

0 

0 

2 

1 

1 

1 

1 

6 

25 

8 

9 

11 

6 

0 

0 

0 

0 

3 

1 

1 

2 

1 

7 

29 

9 

11 

12 

8 

1 

0 

0 

0 

3 

1 

1 

2 

1 

8 

33 

10 

12 

14 

9 

1 

0 

0 

0 

3 

1 

1 

2 

1 

9 

37 

11 

14 

16 

10 

1 

0 

0 

0 

4 

2 

1 

3 

1 

10 

41 

13 

15 

18 

11 

1 

0 

0 

0 

4 

2 

2 

3 

2 

11 

45 

14 

17 

19 

12 

1 

0 

0 

0 

5 

2 

2 

3 

2 

12 

49 

15 

18 

21 

13 

1 

0 

0 

0 

5 

•2 

2 

3 

2 

2 

13 

54 

16 

20 

23 

14 

1 

0 

0 

0 

5 

1 

2 

2 

4 

2 

2 

14 

58 

18 

21 

25 

15 

1 

0 

0 

0 

6 

2 

2 

2 

4 

2 

2 

15 

62 

19 

23 

26 

16 

1 

0 

0 

0 

6 

2 

3 

2 

4 

3 

2 

16 

66 

20 

25 

28 

17 

1 

0 

1 

1 

7 

2 

3 

2 

5 

3 

2 

17 

70 

21 

26 

30 

18 

1 

0 

1 

1 

7 

2 

3 

3 

5 

3 

2 

18 

74 

23 

28 

32 

19 

2 

0 

1 

1 

8 

2 

3 

3 

5 

3 

2 

19 

78 

24 

29 

33 

21 

2 

0 

1 

8 

2 

3 

3 

6 

3 

3 

20 

83 

25 

31   35 

22 

2 

0 

1 

8 

2 

3 

3 

6 

3 

3 

21 

87 

26 

32   37 

23 

2 

0 

1 

9 

2 

4 

3 

6 

4 

3 

22 

91 

28 

34   39 

24 

2 

0 

1 

9 

2 

4 

3 

6 

4 

3 

23 

95 

29 

35   40 

25 

2 

0 

1 

10 

3 

4 

4 

7 

4 

3 

24 

99   31 

37   42 

26  ||  2 

0 

1 

10 

3 

4 

4 

7 

4 

3 

H. 

Sup.  of  Nod. 

II 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

/    r, 

0    ' 

1 

0  7.9 

0  28 

1 

2 

1 

1 

1 

0 

2 

1 

2 

0  15.9 

0  56 

3 

3 

2 

3 

3 

0 

3 

3 

3 

0  23.8 

1  24 

4 

5 

4 

4 

4 

1 

5 

4 

4 

0  31.8 

1  52 

6 

7 

5 

6 

6 

1 

7 

6 

5 

0  39.7 

2  19 

7 

8 

6 

7 

7 

1 

9 

7 

6 

0  47.7 

2  47 

9 

10 

7 

9 

9 

1 

10 

9 

7 

0  55.6 

3  15 

10 

12 

8 

10 

10 

2 

12 

10 

8 

1   3.6 

3  43 

11 

13 

9 

11 

12 

2 

14 

11 

9 

11.5 

4  11 

13 

15 

11 

13 

13 

2 

15 

13 

10 

19.4 

4  39 

14 

16 

12 

14 

15 

2 

17 

14 

11 

27.4 

5  7 

16 

18 

13 

15 

16 

2 

19 

15 

12 

35.3 

5  35 

17 

20 

14 

17 

18 

3 

21 

17 

13 

43.3 

6  2 

18 

21 

15 

18 

19 

3 

23 

18 

14 

51.2 

6  30 

20 

23 

16 

19 

21 

3 

24 

19 

15 

1  59.2 

6  58 

21 

25 

18 

21 

22 

3 

26 

21 

16 

2  7.1 

7  26 

23 

26 

19 

22 

24 

4 

28 

22 

17 

2  15.0 

7  54 

24 

28 

20 

24 

25 

4 

29 

24 

18 

2  23.0 

8  22 

26 

29 

21 

25 

27 

4 

31 

25 

19 

2  30.9 

8  50 

27 

31 

22 

27 

28 

4 

33 

27 

20 

2  38.9 

9  18 

28 

32 

24 

28 

30 

4 

35 

28 

21 

2  46.8 

9  45 

30 

34 

25 

29 

31 

5 

37 

29 

22 

2  54.8 

10  13 

31 

36 

26 

31 

33 

5 

38 

31 

23 

3  2.7 

10  41 

33 

38 

27 

32 

34 

5 

40 

32 

24 

3  10.6 

11  9 

34 

39 

28 

34 

36   5 

42 

34 

TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


c 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

31 

6 

14 

22 

6 

7 

8 

1 

8 

0 

1 

1 

1 

2 

1 

1 

32 

6 

14 

23 

6 

7 

8 

1 

9 

1 

2 

2 

1 

2 

1 

1 

33 

6 

15 

24 

7 

8 

9 

9 

1 

2 

2 

1 

2 

1 

34 

6 

15 

25 

7 

8 

9 

9 

2 

2 

1 

2 

35 

7 

10 

25 

7 

8 

9 

10 

2 

2 

1 

2 

36 

7 

16 

26 

7 

8 

9 

10 

2 

2 

1 

3 

37 

7 

17 

27 

7 

9 

10 

10 

2 

2 

1 

3 

38 

7 

17 

27 

8' 

9 

10 

2 

10 

2 

2 

1 

3 

39 

7 

18 

28 

8 

9 

10 

2 

11 

2 

2 

1 

3 

1 

40 

8 

18 

29 

8 

9 

10 

2 

11 

2 

2 

1 

3 

1 

41 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

1 

3 

1 

42 

8 

19 

30 

8 

10 

11 

2 

11 

2 

2 

1 

3 

43 

8 

19 

31 

9 

10 

11 

2 

12 

2 

2 

1 

3 

44 

8 

20 

32 

9 

10 

11 

2 

12 

2 

2 

1 

3 

45 

9 

20 

32 

9 

10 

12 

2 

12 

2 

2 

1 

3 

46 

9 

21 

33 

9 

11 

12 

2 

12 

2 

2 

1 

3 

47 

9 

21 

34 

9 

11 

12 

2 

13 

2 

2 

1 

3 

48 

9 

22 

35 

10 

11 

12 

2 

13 

2 

2 

1 

3 

49 

9 

22 

35 

10 

11 

13 

2 

13 

2 

2 

1 

3 

50 

9 

23 

36 

10 

11 

13 

2 

13 

2 

2 

1 

3 

51 

10 

23 

37 

10 

12 

13 

2 

14 

2 

2 

1 

4 

52 

10 

24 

38 

10 

12 

13 

2 

14 

3 

3 

1 

4 

53 

10 

24 

38 

11 

12 

14 

2 

14 

3 

3 

1 

4 

54 

10 

24 

39 

11 

12 

i4 

2 

14 

3 

1 

3 

1 

4 

2 

55 

10 

25 

40 

11 

13 

14 

2 

15 

3 

1 

3 

1 

4 

2 

56 

11 

25 

40 

11 

13 

14 

2 

15 

3 

1 

3 

1 

4 

2 

57 

11 

26 

41 

11 

13 

15 

2 

15 

3 

1 

3 

1 

4 

2 

58 

11 

26 

42 

12 

13 

15 

2 

16 

3 

1 

3 

1 

4 

2 

2 

59 

11 

27 

43 

12 

14 

15 

2 

16 

3 

1 

3 

1 

4 

2 

2 

!60 

11 

27 

43 

12 

14 

15 

2 

16 

3 

1 

3 

1 

4  1 

2 

2 

TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


up. 

Min. 

Evec. 

Anom. 

Varia. 

Long. 

Nod. 

II 

V 

VI 

VII 

vm 

IX 

XI 

XII 

1 

0  28 

0  32.7 

0  30 

0  32.9 

0.1 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0  57 

1  5.3 

1  1 

1  5.9 

0.3 

1 

0 

0 

0 

0 

0 

0 

0 

3 

1  25 

1  38.0 

1  31 

1  38.8 

0.4 

1 

0 

0 

0 

0 

0 

0 

0 

4 

1  53 

2  10.6 

2  2 

2  11.8 

0.5 

2 

0 

0 

0 

0 

0 

0 

0 

5 

2  2] 

2  43.3 

2  32 

2  44.7 

0.7 

2 

0 

0 

0 

0 

0 

0 

0 

6 

2  50 

3  16.0 

3  3 

3  17.6 

0.8 

3 

0 

0 

0 

0 

0 

0 

0 

7 

3  18 

3  48.6 

3  33 

3  50.6 

0.9 

3 

0 

0 

0 

0 

0 

0 

0 

8 

3  46 

4  21.3 

4  4 

4  23.5 

1.1 

4 

0 

0 

0 

0 

0 

0 

0 

9 

4  15 

4  54.0 

4  34 

4  56.5 

.2 

4 

0 

0 

0 

0 

0 

0 

0 

10 

4  43 

5  26.6 

5  5 

5  29.4 

.3 

5 

0 

0 

0 

0 

0 

0 

0 

11 

5  11 

5  59.3 

5  35 

6  2.4 

.5 

5 

0 

0 

0 

0 

0 

0 

0 

12 

5  40 

6  31.9 

6  6 

6  35.3 

.6 

6 

0 

0 

0 

0 

0 

0 

0 

13 

6  8 

7  4.6 

6  36 

7  8.2 

.7 

6 

0 

0 

0 

0 

0 

0 

0 

14 

6  36 

7  37.3 

7  7 

7  41.2 

.9 

7 

0 

0 

0 

0 

0 

0 

0 

15 

7  4 

8  9.9 

7  37 

8  14.1 

2.0 

7 

0 

0 

0 

0 

0 

0 

0 

16 

7  33 

8  42.6 

8  8 

8  47.1 

2.1 

7 

0 

0 

0 

0 

0 

0 

0 

17 

8  1 

9  15.3 

8  38 

9  20.0 

2.3 

8 

0 

0 

0 

0 

0 

0 

0 

18 

8  29 

9  47.9 

9  9 

9  52.9 

2.4 

8 

0 

0 

0 

0 

0 

1 

0 

19 

8  58 

10  20.6 

9  39 

10  25.9 

2.5 

9 

0 

0 

0 

0 

0 

1 

0 

20 

9  26 

10  53.2 

10  10 

10  58.8 

2.6 

9 

0 

1 

0 

0 

0 

1 

0 

21 

9  54 

11  25.9 

10  40 

11  31.8 

2.8 

10 

0 

0 

0 

0 

1 

0 

22 

10  22 

11  58.6 

11  11 

12  4.7 

2.9 

10 

1 

0 

0 

1 

0 

23 

10  51 

12  31.2 

11  41 

12  37.6 

3.0 

11 

1 

0 

0 

0 

24 

11  19 

13  3.9 

12  12 

13  10.6 

3.2 

11 

1 

0 

25 

11  47 

13  36.6 

12  42 

13  43.5 

3.3 

12 

1 

0 

26 

'.2  16 

14  9.2 

13  13 

14  16.5 

3.4 

12 

1 

1 

27 

.2  44 

14  41.9 

13  43 

14  49.4 

3.6 

13 

1 

1 

28 

13  12 

15  14.6 

14  13 

15  22.3 

3.7 

13 

1 

1 

29 

13  40 

15  47.2 

14  44 

15  55.3 

3.8 

13 

1 

1 

30 

14  9 

16  19.9 

15  14 

16  28.2 

4.0 

14 

1 

1 

1 

64 


TABLE  XXXIX. 


Moon's  Motions  for  Minutes. 


c 

i 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

i 

0 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

2 

0 

1 

1 

0 

0 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

3 

1 

1 

2 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

4 

1 

2 

3 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

5 

1 

2 

4 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

6 

1 

3 

4 

1 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

7 

1 

3 

5 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

8 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

9 

2 

4 

6 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

0 

0 

0 

0 

10 

2 

5 

7 

2 

2 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

0 

0 

11 

2 

5 

8 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

12 

2 

5 

9 

2 

3 

3 

0 

3 

0 

0 

0 

0 

0 

0 

0 

13 

2 

6 

9 

3 

3 

3 

1 

4 

0 

0 

0 

0 

0 

0 

0 

14 

3 

6 

10 

3 

3 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

15 

3 

7 

11 

3 

3 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

16 

3 

7 

12 

3 

4 

4 

1 

4 

0 

0 

0 

0 

0 

0 

0 

17 

3 

8 

12 

3 

4 

4 

1 

5 

0 

0 

0 

0 

0 

0 

0 

18 

3 

8 

13 

4 

4 

5 

1 

5 

0 

0 

0 

0 

0 

0 

19 

4 

9 

14 

4 

4 

5 

1 

5 

0 

0 

0 

0 

0 

0 

20 

4 

9 

14 

4 

5 

5 

1 

5 

0 

0 

0 

0 

1 

0 

21 

4 

10 

15 

4 

5 

5 

1 

6 

0 

0 

0 

1 

0 

0 

22 

4 

10 

16 

4 

5 

6 

1 

6 

0 

0 

2 

0 

0 

23 

4 

10 

17 

5 

5 

6 

1 

6 

0 

0 

2 

0 

0 

24 

5 

11 

17 

5 

6 

6 

1 

7 

0 

2 

1 

0 

25 

5 

11 

18 

5 

6 

6 

1 

7 

0 

2 

1 

0 

26 

5 

12 

19 

5 

6 

7 

1 

7 

0 

2 

1 

p 

27 

5 

12 

19 

5 

6 

7 

1 

7 

0 

2 

1 

6 

28 

5 

13 

20 

6 

7 

7 

1 

8 

0 

2 

1 

0 

29 

6 

13 

21 

6 

7 

7 

1 

8 

0 

2 

1 

o  i 

30 

6 

14 

22 

6 

7 

8 

1 

8 

0' 

1 

9 

1 

°! 

TABLE  XXXIX. 


65 


Moon's  Motions  for  Minutes. 


Min. 

Evec. 

Anora. 

Varia. 

Long. 

Sup. 
Nod. 

II 

V 

VI 

VII 

vra 

IX 

XI 

s 

31 

14  37 

16  52.5 

15  45 

17  1.2 

4.1 

14 

32 

15  5 

17  25.2 

16  15 

17  34.1 

4.2 

15 

33 

15  34 

17  57.9 

16  46 

18  7.1 

4.4 

15 

34 

16  2 

18  30.5 

17  16 

18  40.0 

4.5 

16 

35 

16  30 

19  3.2 

17  47 

19  12.9 

4.7 

16 

36 

16  58 

19  35.8 

18  17 

19  45.9 

4.8 

17 

37 

17  27 

20  8.5 

18  48 

20  18.8 

4.9 

17 

38 

17  55 

20  41.2 

19  18 

20  51.8 

5.0 

18 

39 

18  23 

21  13.8 

19  49 

21  24.7 

5.2 

18 

40 

18  52 

21  46.5 

20  19 

21  57.6 

5.3 

19 

41 

19  20 

22  19.2 

20  50 

22  30.6 

5.4 

19 

42 

19  48 

22  51.8 

21  20 

23  3.5 

5.6 

20 

43 

20  16 

23  24.5 

21  51 

23  36.5 

5.7 

20 

44 

20  45 

23  57.1 

22  21 

24  9.4 

5.8 

21 

45 

21  13 

24  29.8 

22  52 

24  42.3 

6.0 

21 

46 

21  41 

25  2.5 

23  22 

25  15.3 

6.1 

21 

47 

22  10 

25  35.1 

23  53 

25  48.2 

6.2 

22 

48 

22  38 

26  7.8 

24  23 

26  21.2 

6.4 

22 

49 

23  6 

26  40.5 

24  54 

26  54.1 

6.5 

23 

50 

23  34 

27  1.3.1 

25  24 

27  27.0 

6.6 

23 

51 

24  3 

27  45.8 

25  55 

28  0.0 

6.8 

24 

52 

24  31 

28  18.5 

26  25 

28  32.9 

6.9 

24 

53 

24  59 

28  51.1 

26  56 

29  5.9 

7.0 

25 

54 

25  28 

29  23.8 

27  26 

29  38.8 

7.1 

25 

2 

55 

25  56 

29  56.4 

27  66 

30  11.8 

7.3 

26 

2 

56 

26  24 

30  29  1 

28  27 

30  44.7 

7.4 

26 

1 

2 

57 

26  52 

31  1.8 

28  57 

31  17.6 

7.5 

27 

2 

2 

58 

27  21 

31  34.4 

29  28 

31  506 

7.7 

27 

2 

2 

59 

27  49 

32  7.1 

29  58 

32  2J  .5 

7.8 

28 

2 

2 

60 

28  17 

32  39.8 

f-»0  29 

32  56.5 

7.9 

28 

1 

2 

2 

1 

66 


TABLE  XL. 


Moon's  Motions  for  Seconds. 


i 

Sec. 

Evec. 

Anom. 

Var. 

Long. 

Sec. 

Evec. 

Anom. 

Var. 

Long. 

1 

0 

0.5 

1 

0.5 

31 

15 

16.9 

16 

17.0 

2 

1 

1.1 

1 

1.1 

32 

15 

17.4 

16 

17.6 

3 

1 

1.6 

2 

1.6 

33 

16 

18.0 

17 

18.1 

4 

2 

2.2 

2 

2.2 

34 

16 

18.5 

17 

18.7 

5 

2 

2.7 

3 

2.7 

35 

17 

19.1 

18 

19.2 

6 

3 

3.3 

3 

3.3 

36 

17 

19.6 

18 

19.8 

7 

3 

3.8 

4 

3.8 

37 

18 

20.1 

19 

20.3 

8 

4 

4.3 

4 

4.4 

38 

18 

20.7 

19 

20.9 

9 

4 

4.9 

5 

4.9 

39 

18 

21.2 

20 

21.4 

10 

5 

5.4 

5 

5.5 

40 

19 

21.8 

20 

22.0 

11 

5 

6.0 

6 

6.0 

41 

19 

22.3 

21 

22.5 

12 

6 

6.5 

6 

6.6 

42 

20 

22.9 

21 

23.1 

13 

6 

7.1 

7 

7.1 

43 

20 

23.4 

22 

23.6 

14 

7 

7.6 

7 

7.7 

44 

21 

24.0 

22 

24.2 

15 

7 

8.2 

8 

8.2 

45 

21 

24.5 

23 

24.7 

16 

8 

8.7 

8 

8.8 

46 

22 

25.0 

23 

25.3 

17 

8 

9.2 

9 

9.3 

47 

22 

25.6 

24 

25.8 

18 

9 

9.8 

9 

9.9 

48 

23 

26.1 

24 

26.4 

19 

9 

10.3 

10 

10.4 

49 

23 

26.7 

25 

26.9 

20 

9 

10.9 

10 

11.0 

50 

24 

27.2 

25 

27.4 

21 

10 

11.4 

11 

11.5 

51 

24 

27.8 

26 

28.0 

22 

10 

12.0 

11 

12.1 

52 

25 

28.3 

26 

28.5 

23 

M 

12.5 

12 

12.6 

53 

25 

28.9 

27 

29.1 

24 

11 

13.1 

12 

13.2 

54 

26 

29.4 

27 

29.6 

25 

12 

13.6 

13 

13.7 

55 

26 

29.9 

28 

30.2 

26 

12 

14.1 

13 

14.3 

56 

26 

30.5 

28 

30.7 

27 

13 

14.7 

14 

14.8 

57 

27 

31.0 

29 

31.3 

28 

13 

15.2 

14 

154 

58 

27 

31.6 

29 

31.8 

29 

14 

15.8 

15 

15.9 

5" 

28 

32.1 

30 

32.4 

30 

14 

16.3 

15 

16.5 

60 

28 

32.7 

30 

32.9 

^a  XLI.  67 

First  Equation  of  Moon's  Longitude. — Argument  1. 


Arg. 

1 

Diff. 
for  10 

Arg. 

Diff. 

1   jfor  10 

Arg. 

1 

Diff. 
for  10 

Arg. 

1 

Diff. 
for  10 

0 

12  40.0 

1  »~>  ( 

2500'   40.7;  '' 

5000 

12  40.0 

4   /\/* 

7500 

23  39.3 

0  0*> 

50  12  18.8 
100  11  57.7 
150  11  36.6 
200  11  15.6 
250  10  54.7 

4.%4 

4.22 
4.22 
4.20  i 
4.18 
4.16 

2550J   41.5  "•;£ 
2600;   42  9  JJ-"° 
2650    45.0  JJ-JJ 
2700    47.7  J'J* 
2750    51.0    ' 

5050 
5100 
5150 
5200 
5250 

13  0.3  J'J£ 

•  q  9n  el  4.04 

!2  1°  -y  4.04 

13  40.7 
14  0.9  J-°* 
14  20.9  J-OJ 

7550'23  39.4 
7600*23  38.9 
7650,23  37.7 
7700!23  35.8 
7750|23  33.3 

U.U'v 

o~To 

0.24 
0.38 
0.50 
0.62 

300 

350 
400 
450 
500 

10  33.9 
10  13.2 
9  52.6 
9  32.3 
9  12.1 

4.14 
4.12 
4.06 
4.04 
4.00 

2800 
2850 
2900 
2950 
3000 

55.0 
59.6 
2  4.8 
2  10.7 
2  17.1 

0.92 
1.04 
1.18 
1.28 
1.42 

5300 
5350 
5400 
5450 
5500 

14  40.9 
1  5  0.8 
15  20.5 
15  40.1 
15  59.6 

3.98 
3.94 
3.92 
3.90 
3.84 

7800 
7850 
7900 
7950 
8000 

23  30.2 
23  26.4 
23  22.0 
23  16.9 
23  11.2 

0.76 
0.88 
1.02 
1.14 
1.26 

550 
600 
650 
700 

750 

8  52.1 
8  32.4 
8  13.0 
7  53.8 
7  34.9 

3.94 

3.88 
3.84 

3.70 

3050 
3100 
3150 
3200 
3250 

2  24.2 
2  31.9 
2  40.1 
2  48.9 
2  58.3 

1.54 
1.64 
1.76 
1.88 
1.98 

5550  16  18.8 
5600  16  37.8 
5650  16  56.7 
5700:17  15.3 
5750117  33.6 

3.80 
3.78 
372 
3.66 
3.60 

8050 
8100 
8150 
8200 
8250 

23  4.9 
22  57.9 
22  50.3 
22  42.0 
22  33.2 

.40 
.52 
1.66 
.76 
.90 

800 
850 
900 
950 
1000 

7  16.4 
6  58.2 
6  40.3 
6  22.8 
6  5.7 

364 
3.58 
3.50 
3.42 
3.34 

3300 
3350 
3400 
3450 
3500 

3  8.2 
3  18.7 
3  29.7 
3  41.3 
3  53.4 

2.10 
2.20 
2.32 
2.42 
2.50 

5800  17  51.6 
585018  9.4 
5900J18  26.9 
5950  18  44.0 
6000  19  0.8 

3.56 
3.50 
3.42 
3.36 
3.28 

8300^22  23.7 
835022  13.7 
840022  3.1 
845021  51.9 
850021  40.1 

2.00 
2.12 
2.24 
2.36 
246 

1050 
1  100 
'150 
1  200 
1250 

5  49.0 
5  32.8 
5  17.0 
5   1.6 
4  46.7 

3.24 
3.16 
3.08 
2.98 
2.88 

3550 
3600 
3650 
3700 
3750 

4  5.9 
4  19.0 
4  32.5 
4  46.5 
5  0.9 

2.62 
2.70 
2.80 
2.88 
2.98 

6050 
6100 
6150 
6200 
6250 

19  17.2 
19  33.3 
19  49.0 
20  4.2 
20  19.1 

3.22 
3.14 
3.04 
2.98 
2.88 

8550)21  278 
860021  15.0 
865021   1.6 
8700  20  47.7 
875020  33.3 

2.56 
268 
2.78 
2.88 
2.98 

1300 
1350 
1400 
1450 
1500 

4  32.3 
4  18.4 
4  5.0 
3  52.2 
3  39.9 

2.78 
2.68 
2.56 
2.46 
2.36 

3800 
3850 
3900 
3950 
4000 

5  15.8 
5  31.0 
5  46.7 
6  2.8 
6  19.2 

3.04 
3.14 
3.22 
3.28 
3.36 

6300 
6350 
6400 
6450 
6500 

20  33.5 
20  47.5 
21   1.0 
21  14.1 
21  26.6 

2.80 
2.70 
2.62 
2.50 
2.42 

880020  18.4 
885020  3.0 
8900  19  47.2 
895019  31.0 
9000(19  14.3 

3.08 
3.16 
3.24 
3.34 
3.42 

1550 
1600 
1650 
1700 
1750 

3  28.1 
3  16.9 
3  6.3 
2  56.3 
2  46.8 

2.24 
2.12 
2.00 
1.90 
1.76 

4050 
4100 
4150 
4200 
4250 

6  36.0 
6  53.1 
7  10.6 

7  28.4 
7  46.4 

3.42 
3.50 
3.56 
3.60 
3.66 

655021  38.7 
6600I21  50.3 
665022  1.3 
670022  11.8 
G75022  21.7 

2.32 
2.20 
2.10 
1.98 
1.88 

9050 
9100 
9150 
9200 
9250 

18  57.2 
18  39.7 
18  21.8 
18  3.6 
17  45.1 

3.50 
3.58 
3.64 
3.70 
3.78 

1800  2  38.0 

cc 

4300!  8  4.7 

680022  31.1 

9300 

17  26.2 

1850  2  29.7 

-DO 

52 

4350  8  23.3  ^ 

6850  22  39.9 

fid 

9350 

17  7.0 

3.84 

q  QQ 

1900  2  22.1 

4-fl 

4400  8  42.2  J2 

690022  48.1 

.0^ 

9400 

,16  47.6 

O.oCJ 

q  QA 

1950 

2  15.1 

.ftU 

26 

4450 

9  1.SJJJ 

695022  55.8 

9450 

16  27.9 

o.yt 

2000 

2  8.8 

.14 

4500 

••HJS 

7000|23  2.9 

!28 

9500 

,16  7.9 

4^04 

2050 
2100 
2150 
2200 
2250 

2  3.1 
1  58.0 
1  53.6 
1  49.8 
1  46.7 

1.02 
0.88 
1  0.76 
0.62 
0.50 

4550  9  39.9 
4600  9  59.5 
465010  19.2 
4700  10  39.1 
4750  10  59.1 

3.92 
3.94 
3.98 
4.00 
4.00 

705023  9.3 
710023  15.2 
715023  20.4 
720023  25.0 
725023  29.0 

1.18 
1.04 
0.92 
0.80 
0.66 

9550 
9600 
9650 
9700 
9750 

|l5  47.7 
15  27.4 
15  6.8 
14  46.1 
14  25.3 

4.06 
4.12 
4.14 
4.16 
4.18 

2300  1  44.2  ft(,( 
2350  1  42.3  JJJ 

4800  11  19.1  Aft. 
4850  11  39.3  *"t 

730023  32.3 
(735023  35.0 

0.54 

9800 
9850 

14  4.4 
13  43.4 

4.20 

2400  1  41.1  JJ 
2450  1  40  6 
2500  1  40.7  0.02 

490011  59.5!  J-Jr:  740023  37.1 
495012  19.7  *-"*b745023  38.5 
5000  12  40.0     S7500  23  39.3 

0.42 
0.28 
0.16 

9900  13  22.3 
0950  13  1.2 
10000  12  40.0 

4.22 
4.22 
4.24 

68  TABLE  XLII. 

Equations  2  to  7  of  Moon's  Longitude.     Arguments  2.  to  7 


Arg. 

2 

diff 

3 

diff|    4 

diff 

5 

diff 

6 

diff 

7 

cliff  I  Arg. 

2500 
2600 
2700 
2800 
2900 
3000 

457.3 
457.0 
456.1 
454.7 
452.7 
450.1 

0.3 
0.9 
1.4 
2.0 
2.6 

0    2.3 
0   2.4 
0   2.8 
0    3.3 
0    4.1 
0    5.1 

0.1 

0.4 
0.5 
0.8 
1.0 

630.3 
629.9 
628.8 
626.9 
624.3 
621.0 

0.4 
1.1 
1.9 
2.6 
3.3 

339.4 
339.2 
338.5 
3  37.5 
336.0 
334.1 

0.2 
0.7 
1.0 
1.5 
1.9 

0    6.2 
0   6.4 
0   6.9 

0    7.7 
0    8.8 
010.3 

0.2 
0.5 
0.8 
1.1 
1.5 

0   0.8 
0   0.9 
0    1.3 
0    1.8 
0    2.7 
0   3.7 

0.1 
0.4 
0.5 
0.9 
1.0 

2500 
2400 
2300 
2200 
2100 
2000 

3.1 

1.3 

4.1 

2.4 

1.8 

1.3 

3100 

447.0 

3  7 

0   6.4 

1.4 

616.9 

4.7 

331.7 

2  7 

012.1 

2.1 

0   5.0 

1.4 

1900 

3200 

443.3 

A    O 

0    7.8 

1.6 

612.2 

K  A 

329.0 

014.2 

2  4 

0    6.4 

1.7 

1800 

3300 

439.1 

%*i6 

A  7 

0    9.4 

1.9 

6    6.8 

V.Tt 

ft    1 

3  25.9 

3  5 

016.6 

2.6 

0    8.1 

1.9 

1700 

3400 
3500 

434.4 
4292 

TC.  1 

5.2 

011.3 
013.3 

2^0 

6    0.7 
554.0 

0.  1 

6.7 

322.4 
318.5 

3^9 

019.2 
022.2 

3.0 

010.0 
012.1 

2.1 

1600 
1500 

5.7 

2.2 

7.4 

4.2 

3.2 

2.3 

3600 

4235 

6.1 

0  15.5 

2.4 

546.6 

7.9 

314.3 

4.6 

025.4 

3.5 

014.4 

2.4 

1400 

3700 

4174 

6.6 

0  17.9 

2.6 

538.7 

o  4 

3    9.7 

4.8 

028.9 

3.8 

016.8 

2.7 

1300 

3800 
3900 

4  10.8 
4    3.9 

6.9 
7.3 

020.5 
023.2 

2.7 
2.9 

530.3 
521.3 

O.*r 

9.0 
9.4 

3    4.9 
259.7 

5^2 

5.4 

032.7 
036.6 

3^9 
4.1 

0  19.5 
022.3 

2.8 
2.9 

1200 
1100 

4000 

356.6 

7.7 

026.1 

3.0 

511.9 

9^9 

2543 

5.7 

040.7 

4.4 

025.2 

3.1 

1000 

4100 

348.9 

7.9 

029.1 

3.1 

5    2.0 

10.3 

2  48.  6 

5.9 

045.1 

4.5 

0283 

3.2 

900 

4200 

341.0 

8.3 

032.2 

3.2 

451.7 

10.7 

242  / 

6.1 

049.6 

4.7 

031  5 

3.3 

800 

4300 
4400 

3  32.7 
324.2 

8.5 
8.7 

035.4 
038.8 

3.4 
3.4 

441.0 
430.1 

10^9 
11  3 

236.6 
2  30.3 

6.3 
6.5 

0  54.3 
059.2 

4.9 
4.9 

0348 
038.2 

3.4 
3.5 

700 
600 

4500 

3  15.5 

8.9 

042.2 

3.5 

418.8 

11.5 

223.8 

6.6 

1    4.1 

5.1 

041.7 

3.6 

500 

4600 

3    6.6 

9.0 

045.7 

3.5 

4   7.3 

11.6 

217.2 

6.7 

1    9.2 

5.1 

045.3 

3.6 

400 

4700 

257.6 

9.1 

049.2 

3.6 

355.7 

11.8 

2  10.5 

6.8 

1143 

048.9 

3.7 

300 

4800 

248.5 

9.3 

052.8 

3.6 

343.9 

12.0 

2    3.7 

6.8 

1  19.5 

52 

052.6 

3.7 

200 

4900 

239.2 

9.2 

0  56.4 

3.6 

331.9 

Uq 

56.9 

6.9 

124.7 

5.3 

056.3 

3.7 

100 

5000 

230.0 

9.2 

1    0.0 

3.6 

320.0 

.»/ 
11.9 

50.0 

6.9 

1  30.0 

5.3 

1    0.0 

3.7 

0 

5100 

220.8 

9.3 

3.6 

3.6 

3    8.1 

12.0 

43  1 

6.8 

135.3 

5.2 

3.7 

3.7 

9900 

5200 

211.5 

9.1 

7.2 

3.6 

256.1 

11  8 

36.3 

6.8 

1  40.5 

5.2 

7.4 

3.7 

9800 

5300 

2   2.4 

9.0 

1  10.8 

3.5 

244.3 

11.6 

29.5 

6.7 

145.7 

5.1 

11.1 

3.6 

9700 

5400 

153.4 

8.9 

14.3 

3.5 

232.7 

11.5 

22.8 

6.6 

150.8 

147 

3.6 

9600 

5500 

144.5 

8.7 

17.8 

3.4 

221.2 

11.3 

16.2 

6.5 

155.9 

4.9 

1  183 

3.5 

9500 

5600 

135.8 

8.5 

21.2 

3.4 

2    9.9 

10.9 

9.7 

6.3 

2    08 

4.9 

121.8 

3.4 

9400 

5700 
5800 
5900 
J6000 

127.3 
1  19.0 
1  11.1 
1    3.4 

8.3 
7.9 
7.7 
7.3 

24.6 
27.8 
30.9 
33.9 

3.2 
3.1 
3.0 
2.9 

1  59.0 
148.3 
138.0 
128.1 

10.7 
10.3 
9.9 
9.4 

1    3.4 
057.3 
051.4 
045.7 

6.1 
5.9 
5.7 
5.4 

2    5.7 
210.4 
214.9 
219.3 

4^7 
4.5 
4.4 
4.1 

125.2 
28.5 
31.7 
134.8 

3.3 
3.2 
3.1 
2.9 

9300 
9200 
9100 
9000 

6100 
;6200 
J6300 
6400 
6500 

056.1 
049.2 
042.6 
036.5 
030.8 

6.9 
6.6 
6.1 
5.7 

36.8 
39.5 
42.1 
44.5 
46.7 

2.7 
2.6 
2.4 
2.2 

118.7 
1    9.7 
1    1.3 
053.4 
046.0 

9.0 

8.4 
7.9 
7.4 

0403 
035.1 
030.3 
025.7 
021.5 

5.2 
4.8 
4.6 
4.2 

223.4 
2  27.3 
231.1 
234.6 
237.8 

3.9 
3.8 
3.5 
3.2 

137.7 
140.5 
143.2 
145.6 
147.9 

2.8 
2.7 
2.4 
2.3 

8900 
8800 
8700 
8600 
8500 

5.2 

2.0 

6.7 

3.9 

3.0 

2.1 

6600 
6700 
6800 
6900 
7000 

025.6 
020.9 
016.7 
013.0 
0    9.9 

47 
4.2 
37 
3.1 

148.7 
1  50.6 
152.2 
153.6 
154.9 

.9 
.6 
.4 
.3 

039.3 
033.2 

027.8 
023.1 
019.0 

6.1 

5.4 
4.7 
4.1 

017.6 
014.1 
011.0 
0    8.3 
0    5.9 

3.5 
3.1 
2.7 
2.4 

240.8 
243.4 
V  45.8 
247.9 
249.7 

2.6 
2.4 

2.1 
1.8 

1  50.0 
151.9 
1  53.6 
155.0 
1563 

1.9 
1.7 
1.4 
1.3 

8400 
8300 
8200 
8100 
8000 

2.6 

.0 

3.3 

1.9 

1.5 

1.0 

7100 
7200 
7300 
7400 
7500 

0    7.3 
0    5.3 
0    3.9 
0   3.0 
0    2.7 

2.0 
1.4 
0.9 
0.3 

155.9 
156.7 
157.2 
157.6 
157.7 

0.8 
05 
0.4 
0.1 

015.7 
013.1 
011.2 
010.1 
0   9.7 

2.6 
1.9 
1.1 
0.4 

0   4.0 

0   2.5 
0    1.5 
0   0.8 
0   06 

1.5 
1.0 
0.7 
0.2 

251.2 
252.3 
2  53.  1 
253.6 
253.8 

1.1 
0.8 
0.5 
0.2 

157.3 
1  58.2 
158.7 
159.1 
159.2 

0.9 
0.5 
0.4 
0.1 

7900 
7800 
7700 
7600 
7500 

TABLE  XLIII. 


TABLE  XL1V.       69 


Equations  8  and  9. 


Equations  10  and  11. 


Arg. 

8 

9 

Arg. 

8 

9 

0 

1  20.0 

1  200 

5000 

1  20.0 

1  20.0 

100 

1  15.5 

1  287 

5100 

1  24.4 

1  25.8 

200 

I  11.1 

1  37.3 

5200 

1  28.8 

1  31.4 

300 

1  6.7 

1  45.7 

5300 

1  33.1 

36.9 

400 

1  2.3 

1  53.7 

5400 

1  37.4 

42.0 

500 

0  58.0 

2  1.3 

5500 

1  41.6 

46.8 

600 

0  53.8  2  8.3 

5600 

1  45.8 

51.0 

700 

0  49.7 

2  14.7 

5700 

1  49.8 

54.6 

800 

0  45.7 

2  20.2 

5800 

1  53.8 

57.6 

900 

0  41.9 

2  25.0 

5900 

1  57.6 

1  59.8 

1000 

0  38.2 

2  28.9 

6000 

2  1.2 

2  1.3 

1100 

0  34.7  2  31.9 

6100 

2  4.7 

2  1.9 

1200 

0  31.4  j  2  33.9 

6200 

2  8.0  2  1.7 

1300 

0  28.2 

2  34.9 

6300 

2  11.2 

2  0.7 

1400 

0  25.3 

2  35.0 

6400 

2  14.1 

1  58.8 

1500 

0  22.6 

2  34  1 

6500 

2  16.8 

1  56.1 

1600 

0  20.1 

2  322 

6600 

2  19.3 

1  52.5 

1700 

0  17.9 

2  29.5 

6700 

2  21.6 

1  48.3 

1800 

0  15.9 

2  25.9 

6800 

2  23.7 

1  43.4 

1900 

0  14.2 

2  21.5 

6900 

2  25.4 

1  37.8 

2000 

0  12.7 

2  16.4 

7000 

2  27.0 

1  31.7 

2100 

0  11.5 

2  10.7 

7100 

2  28.2 

1  25.1 

2200 

0  10.5 

2  4.4 

7200 

2  29.2 

1  18.2 

2300 

0  9.9 

1  57.7 

7300 

2  30.0 

1  11.1 

2400 

0  9.5 

1  50.7 

7400 

2  30.4 

1  3.8 

2500 

0  9.4 

1  43.5 

7500 

2  30.6 

0  56.5 

2600 

0  9.6 

1  36.2 

7600 

2  30.5 

0  49.3 

2700 

0  10.1 

1  28.9 

7700 

2  30.1 

0  42.3 

2800 

0  10.8 

1  21.8 

7800 

2  29.5 

0  35.6 

2900 

0  11.8 

1  14.9 

7900 

2  28.5 

0  29.3 

3000 

0  13.0 

1  8.3 

8000 

2  27.3 

0  23.6 

3100 

0  14.6 

1  2.2 

8100 

2  25.8  0  18.5 

3200 

0  16.3 

0  56.6 

8200 

2  24.1  0  14.1 

3300 

0  18.4 

0  51.7 

8300 

2  22.1  0  10.5 

3400 

0  20.7 

0  47.5 

8400 

2  19.9 

0  7.8 

3500 

0  23.2 

0  43.9 

8500 

2  17.4 

0  5.9 

3600 

0  25.9 

0  41.2 

8600 

2  14.7  JO  5.0 

3700 

0  28.8 

0  39.3 

8700 

2  11.810  5.1 

3800 

0  32.0 

0  38.3 

8800 

2  8.6 

0  6.1 

3900 

0  35.3 

0  38.1 

8900 

2  5.3 

0  8.1 

4000 

0  38.8 

0  38.7 

9000 

2  1.8 

0  11.1 

410010  42.4  !0  40.2 

9100 

1  58.1 

0  15.0 

4200 

0  46.2 

0  42.4 

9200 

1  54.3 

0  19.8 

4300 

0  50.2 

0  45.4 

9300 

1  50.3 

0  25.3 

4400 

0  54.2 

0  49.0 

9400 

1  46.2 

0  31.7 

4500 

0  58.4 

0  53.2 

9500 

1  42.0 

0  38:7 

4600 

1  2.6 

0  58.0 

9600 

1  37.7 

0  46.3 

4700 

1  6.9 

1  3.1 

9700 

1  33  3 

0  54.3 

4800 

1  11.2 

1  8.6 

9800 

1  28.9 

1  2.7 

4900 

1  15.6 

1  14.2 

9900 

1  24.5 

1  11.3 

6000 

1  20.0 

1  20.0 

1000011  20.0 

1  20.0 

Arg. 

10 

11 

Arg. 

10 

11 

0 

10.0 

10.0 

500 

10.0 

10.0 

10 

9.3 

11.1 

510 

9.6 

10.8 

20 
30 

8.6 
8.0 

12.1 
13.1 

520 
530 

9.2 
8.9 

11.5 
12.3 

40 

7.4 

14.1 

540 

8.5 

12.9 

50 

6.8 

15.0 

550 

8.2 

13.6 

60 

6.2 

15.8 

560 

7.9 

14.2 

70 

5.7 

16.6 

570 

7.7 

14.6 

80 

5.3 

17.3 

580 

7.5 

15.0 

90 

4.9 

17.9 

590 

7.4 

15.4 

100 

4.6 

18.3 

600 

7.3 

15.6 

110 

4.3 

18.6 

610 

7.2 

15.7 

120 

4.1 

18.9 

620 

7.3 

15.7 

130 

4.0 

19.0 

630 

7.4 

15.6 

140 

4.0 

18.9  640 

7.5 

15.4 

150 

4.0 

18.8]  650 

7.8 

15.1 

160 

4.2 

18.6 

660 

8.1 

14.7 

170 

4.4 

18.2 

670 

8.4 

14.2 

180 

4.6 

17.7 

680 

8.7 

13.5 

190 

4.9 

17.1 

690 

9.2 

12.8 

200 

5.3 

16.5 

700 

9.7 

12.1 

210 

5.7 

15.7 

710 

10.2 

11.3 

220 

6.2 

14.9 

720 

10.7 

10.4 

230 

6.7 

14.1 

730 

11.2 

9.5 

240 

7.2 

13.2 

740 

11.7 

8.6 

250 

7.7 

12.3 

750 

12.3 

7.7 

260 

83 

11.4 

760 

12.8 

6.8 

270 

8.8 

10.5 

770 

13.3 

5.9 

280 

9.3 

9.6 

780 

13.8 

5.1 

290 

9.8 

8.7 

790 

14.3 

4.3 

300 

10.3 

7.9 

800 

14.7 

3.5 

310 

10.8 

7.2 

810 

15.1 

2.9 

320 

11.3 

6.5 

820 

15.4 

2.3 

330 

11.6 

5.8 

830 

15.6 

1.8 

340 

11.9 

5.3 

840 

15.8 

1.4 

350 

12.2 

4.9 

850 

16.0 

1.2 

360 

12.5 

4.6 

860 

16.0 

1.1 

370 

12.6 

4.4 

870 

16.0 

1.0 

380 

12.7 

4.3 

880 

15.9 

1.1 

390 

12.8 

4.3 

890 

15.7 

1.4 

400 

12.7 

4.4 

900 

15.4 

1.7 

410 

12.6 

4.6 

910 

15.1 

2.1 

420 

12.5 

5.0 

920 

14.7 

2.7 

430 

12.3 

5.4 

930 

14.3 

3.4 

440 

12.1 

5.8 

940 

13.8 

4.2 

450 

11.8 

6.4 

950 

13.2 

5.0 

460 

11.5 

7.1 

960 

12.6  '  5.9 

470 

11.1 

7.7  :  970 

12.0   6.9 

480 

10.8 

8.5  i  980 

11.4   7.9 

490 

10.4 

9.2  910 

10.7   8.9 

500  1  10.0 

10.0  '10(  0 

LOO  LO.O 

33 


70  TABLE  XLV. 

Equations  12  to  19- 


TABLE  XLVL 
Equation  20. 


Arg. 

12 

13 

14 

15 

16 

17 

18 

19 

Arg. 

250 

2.3 

1.6 

7.8 

0.0 

33.7 

3.4 

16.7 

0.4 

250 

260 

2.3 

1.6 

7.8 

0.0 

33.7 

3.4 

16.7 

0.4 

240 

270 

2.4 

1.7 

7.9 

0.1 

33.6 

3.5 

16.6 

0.4 

230 

280 

2.6 

1.9 

8.0 

0.2 

33.5 

3.5 

16.6 

0.5 

220 

290 

2.9 

2.2 

8.2 

0.3 

33.2 

3.6 

16.5 

0.5 

210 

300 

3.2 

2.5 

8.4 

0.5 

33.0 

3.7 

16.4 

0.6 

200 

310 

3.5 

2.9 

8.7 

0.7 

32.7 

3.9 

16.2 

0.7 

190 

320 

4.0 

3.4 

9.0 

1.0 

32.4 

4.0 

16.1 

0.8 

180 

330 

4.5 

3.9 

9.3 

1.2 

32.0 

4.2 

15.9 

.0 

170 

340 

5.1 

4.4 

9.7 

1.6 

31.6 

4.4 

15.7 

.1 

160 

350 

5.7 

5.1 

10.1 

1.9 

31.1 

4.7 

15.4 

.3 

150 

360 

6.4 

5.8 

10.6 

2.3 

30.6 

4.9 

15.2 

.5 

140 

370 

7.1 

6.6 

11.1 

2.7 

30.1 

5.2 

14.9 

.7 

130 

380 

7.9 

7.4 

11.7 

3.2 

29.4 

5.5 

14.6 

.9 

120 

390 

8.7 

8.3 

12.2 

3.6 

28.7 

5.8 

14.3 

2.1 

110 

400 

9.6 

9.2 

12.8 

4.1 

28.0 

6.1 

13.9 

2.3 

100 

410 

10.5 

10.1 

13.5 

4.6 

27.3 

6.5 

13.6 

2.5 

90 

420 

11.5 

11.1 

14.1 

5.2 

26.6 

6.8 

13.2 

2.8 

80 

430 

12.5 

12.2 

14.8 

5.7 

25.8 

7.2 

12.9 

3.1 

70 

440 

13.5 

13.2 

15.5 

6.3 

25.0 

7.6 

12.5 

3.3 

60 

450 

14.5 

14.3 

16.2 

6.9 

24.2 

8.0 

12.1 

3.6 

50 

460 

15.6 

15.4 

17.0 

7.5 

23.4 

8.4 

11.7 

3.9 

40 

470 

16.7 

16.5 

17.7 

8.1 

22.6 

8.8 

11.3 

4.1 

30 

480 

17.8 

17.7 

18.5 

8.7 

21.7 

9.2 

10.8 

4.4 

20 

490 

18.9 

18.8 

19.2 

9.4 

20.9 

9.6 

10.4 

4.7 

10 

500 

20.0 

20.0 

20.0 

10.0 

20.0 

10.0 

10.0 

5.0 

0 

510 

21.1 

21.2 

20.8 

10.6 

19.1 

10.4 

9.6 

5.3 

990 

520 

22.2 

22.3 

21.5 

11.3 

18.3 

10.8 

9.2 

5.6 

980 

530 

23.3 

23.5 

22.3 

11.9 

17.4 

11.2 

8.7 

5.9 

970 

540 

24.4 

24.6 

23.0 

12.5 

16.6 

11.6 

8.3 

6.1 

960 

550 

25.5 

25.7 

23.8 

13.1 

15.8 

12.0 

7.9 

6.4 

950 

560 

26.5 

26.8 

24.5 

13.7 

15.0 

12.4 

7.5 

6.7 

9iO 

570  I  27.5 

27.8 

25.2 

14.3 

14.2 

12.8 

7.1 

6.9 

930 

580 

28.5 

28.9 

25.9 

14.8 

13.4 

13.2 

6.8 

7.2 

920 

590 

29.5 

29.9 

26.5 

15.4 

12.7 

13.5 

6.4 

7.5 

910 

600 

30.4 

30.8 

27.2 

15.9 

12.0 

13.9 

6.1 

7.7 

900 

610 

31.3 

31.7 

27.8 

16.4 

11.3 

14.2 

5.7 

7.9 

890 

620 

32.1 

32.6 

28.3 

16.8 

10.6 

14.5 

5.4 

8.1 

880 

630 

32.9 

33.4 

28.9 

17.3 

9.9 

14.8 

5.1 

8.3 

870 

640 

33.6 

34.2 

29.4 

17.7 

9.4 

15.1 

4.8 

8.5 

860 

650 

34.3 

34.9 

29.9 

18.1 

8.9 

15.3 

4.6 

8.7 

850 

660 

34.9 

35.6 

30.3 

18.4 

8.4 

15.6 

4.3 

8.9 

840 

670 

35.5 

36.1 

30.7 

18.8 

8.0 

15.8 

4.1 

9.0 

830 

680 

36.0 

36.6 

31.0 

19.0 

7.6 

16.0 

3.9 

9.2 

820 

690 

36.5 

37.1 

31.3 

19.3 

7.3 

16.1 

3.8 

9.3 

810 

700 

36.8 

37.5 

31.6 

19.5 

7.0 

16.3 

3.6 

9.4 

800 

710 

37.1 

37.8 

31.8 

19.7 

6.8 

16.4 

3.5 

9.5 

790 

72  0 

37.4 

38.1 

32.0 

19.8 

6.5 

16.5  3.4 

9.5 

780 

730 

37.6 

38.3 

32.1 

19.9 

6.4  1  16.5  3.4 

9.6 

770 

740 

37.7  38.4 

32.2 

20.0 

6.3 

16.6  3.3 

9.6 

760 

[TOO 

37.7  38.4 

32.2 

20.0 

6.3 

16.6  3.3 

9.6 

750 

Arg. 

20 

Arg. 

0 

10.0 

500 

10 

10.9 

510 

20 

11.8 

520 

30 

12.7 

530 

40 

13.5 

540 

50 

14* 

550 

60 

15.0 

560 

70 

15.7 

570 

80 

16.2 

580 

90 

16.7 

590 

100 

17.0 

600 

110 

17.2 

610 

120 

17.4 

620 

130 

17.4 

630 

140 

17.2 

640 

150 

17.0 

650 

160 

16.7 

660 

170 

16.2 

670 

180 

15.7 

680 

190 

15.0 

690 

200 

14.3 

700 

210 

13.5 

710 

220 

12.7 

720 

230 

11.8 

730 

240 

10.9 

740 

250 

10.0 

750 

260 

9.1 

760 

270 

8.2 

770 

280 

7.3 

780 

290 

6.5 

790 

300 

5.7 

800 

310 

5.0 

810 

320 

4.3 

820 

330 

3.8 

830 

340 

3.3 

840 

350 

3.0 

850 

360 

2.8 

860 

370 

2.6 

870 

380 

2.6 

880 

390 

2.8 

890 

400 

3.0 

900 

410 

3.3 

910 

420 

3.8 

920 

430 

4.3 

930 

440 

5.0 

940 

450 

5.7 

950 

460 

6.5 

960 

470 

7.3 

970 

480 

8.2 

980 

490 

9.1 

990 

[500 

10.0 

1000 

TABLE  XLVII. 


Equations  21  to  29. 


TABLE  XLVIIL  71 
Equations  30  and  31 


1 

21 

22 

23 

24 

25 

26 

27 

28 

29 

> 

03 

25 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

25 

27 

7.8 

3.2 

7.1 

6.1 

5.9 

4.1 

5.8 

4.3 

5.7 

23 

29 

7.7 

3.3 

7.0 

6.1 

5.9 

1.1 

5.8 

4.3 

5.7 

21 

31 

7.6 

3.3 

7.0 

6.0 

5.8 

4.2 

5.7 

4.3 

5.7 

19 

33 

7.5 

3.4 

6.8 

6.0 

5.8 

4.2 

5.7 

4.4 

5.6 

17 

35 

7.3 

3.5 

67 

5.9 

5.7 

4.3 

5.6 

4.4 

5.6 

15 

37 

7.0 

3.7 

6.5 

5.8 

5.7 

4.3 

5.6 

4.5 

5.5 

13 

39 

6.8 

3.9 

6.3 

5.7 

5.6 

4.4 

5.5 

4.6 

5.4 

11 

41 

6.5 

4.0 

6.1 

5.6 

5.5 

4.5 

5.4 

4.6 

5.4 

09 

43 

6.2 

4.2 

5.9 

5.5 

5.4 

4.6 

5.3 

4.7 

5.3 

07 

45 

5.9 

4.4 

5.6 

5.3 

5.3 

4.7 

5.2 

4.8 

5.2 

05 

147 

5.5 

4.7 

5.4 

5.2 

5.2 

4.8 

5.1 

4.9 

5.1 

03 

49 

5.2 

4.9 

5.1 

5.1 

5.1 

4.9 

5.0 

5.0 

5.0 

01. 

51 

4.8 

5.1 

4.9 

4.9 

4.9 

5.1 

5.0 

5.0 

5.0 

99 

53 

4.5 

5.3 

4.6 

4.8 

4.8 

5.2 

4.9 

5.1 

4.9 

97 

55 

4.1 

5.6 

4.4 

4.7 

4.7 

5.3 

4.8 

5.2 

4.8 

95 

57 

3.8 

5.8 

4.1 

4.5 

4.6 

5.4 

4.7 

5.3 

4.7 

93 

59 

3.5 

6.0 

3.9 

4.4 

4.5 

5.5 

4.6 

5.4  ' 

4.6 

91 

61 

3.2 

6.1 

3.7 

4.3 

4.4 

5.6 

4.5 

5.4 

4.6 

89 

63 

3.0 

6.3 

3.5 

4.2 

4.3 

5.7 

4.4 

5.5  j 

4.5 

87 

65 

2.7 

6.5 

3.3 

4.1 

4.3 

5.7 

4.4 

5.6 

4.4 

85 

67 

2.5 

6.6 

3.2 

4.0 

4.2 

5.8 

4.3 

5.6 

4.4 

83 

69 

2.4 

6.7 

3.0 

4.0 

4.2 

5.8 

4.3 

5.7 

4.3 

81 

71 

2.3 

6.7 

3.0 

3.9 

4^1 

5.9 

4.2 

5.7 

4.3 

79 

73 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7  1 

4.3 

77 

75 

2.2 

6.8 

2.9 

3.9 

4.1 

5.9 

4.2 

5.7  J 

4.3  1 

75 

TABLE  XLIX. 
Equation  32.     Argument,  Supp.  of  Node. 


Ills 

IV* 

V* 

VI* 

VII* 

VIII* 

o 

// 

» 

// 

// 

// 

« 

0 

0 

3.1 

4.0 

6.5 

10.0 

13.5 

16.0 

30 

2 

3.1 

4.2 

6.8 

10.2 

13.7 

16.1 

28 

4 

3.1 

4.3 

7.0 

10.5 

13.8 

16.2 

26 

6 

3.1 

4.4 

7.2 

10.7 

14,0 

16.3 

24 

8 

3.2 

4.6 

7.4 

11.0 

14.2 

16.4 

22 

10 

3.2 

4.7 

7.fi 

11.2 

14.4 

16.5 

20 

12 

3.3 

4.9 

7.9 

11.4 

14.6 

16.6 

18 

14 

3.3 

5.0 

8.1 

11.7 

14.8 

16.6 

16 

16 

3.4 

5.2 

8.3 

11.9 

15.0 

16.7 

14 

18 

3.4 

5.4 

8.6 

12.1 

15.1 

16.7 

12 

20 

3.5 

5.6 

8.8 

12.4 

15.3 

16.8 

10 

22 

3.6 

5.8 

9.0 

12.6 

15.4 

16.8 

8 

24 

3.7 

6.0 

9.3 

12.8 

15.6 

16.9 

6 

26 

3.8 

6.2 

9.5 

13.0 

15.7 

16.9 

4 

28 

3.9 

6.3 

9.8 

13.2 

15.8 

16.9 

2 

30 

4.0 

6.5 

10.0 

13.5 

16.0 

16.9 

0 

II* 

10 

O* 

XI* 

X* 

IX* 

AT 

30 

31 

( 

6.0 

5.0 

< 

5.0 

5.0 

t 

4.9 

5.1 

( 

4.9 

5.1 

8 

4.8 

5.2 

10 

4.8 

5.2 

12 

4.7 

5.3 

14 

4.6 

5.4 

16 

4.5 

5.5 

18 

4.4 

5.5 

20 

4.2 

5.6 

22 

4.1 

5.7 

24 

4.0 

5.8 

26 

3.9 

5.8 

28 

3.8 

5.9 

30 

3.7 

5.9 

32 

3.7 

5.9 

34 

3.7 

5.9 

36 

3.7 

5.9 

38 

3.8 

5.8 

40 

3.9 

5.7 

42 

4.1 

5.6 

44 

4.3 

5.5 

46 

4.5 

5.3 

48 

4.8 

5.2 

50 

5.0 

5.0 

52 

5.2 

4.8 

54 

5.5 

4.7 

56 

5.7 

4.5 

58 

5.9 

4.4 

60 

6.1 

4.3 

62 

6.2 

4.2 

64 

6.3 

4.1 

66 

6.3 

4.1 

68 

6.3 

4.1 

70 

6.3 

4.1 

72 

6.2 

4.1 

74 

6.2 

4.2 

76 

6.0 

4.2 

78 

5.9 

4.3 

80 

5.8 

4.4 

82 

5.7 

4.5 

84 

5.5 

4.6 

86 

5.4 

4.6 

88 

5.3 

4.7 

90 

5.2 

4.8 

92 

.1 

48 

94 

.1 

49 

96 

.0 

4.9 

98 

.0 

5.0 

100 

.0  I 

5.0 

Constant  55" 


TABLE   L. 
Evection. 

Argument.     Evection,    corrected. 


0» 

I* 

II* 

Ills 

IV, 

i 

1 

1° 

Diff 

2o 

Diff.  2° 

Diff. 

2° 

Diff 

2o 

Diff 

2o 

Diff. 

(J 

1 

2 
3 
4 
5 

30  00.0 
31  25.5 
32  50.9 
34  16.3 
3541.6 
37    6.7 

85.5 
85.4 
85.4 
85.3 
85.1 

10  43.5 
11  56.7 
13    9.0 
14  20.6 
1531.3 
1641.1 

73.2 
72.3 
71.6 
70.7 
69.8 

40    9.7 
40  50.6 
41  30.1 
42    8.3 
4245.1 
43  20.6 

40.9 
39.5 
38.2 
36.8 
35.5 

50  25.5 
50  23.5 
5020.1 
50  15.2 
50    8.8 
50    1.0 

2.0 
3.4 
4.9 
6.4 
7.8 

39  8.3 
38  24.9 
37  40.4 
36  54.6 
36  7.6 
35  19.5 

43.4 
44.5 
45.8 
47.0 
48.1 

942.0 
829.3 
7  16.0 
6    2.0 
447.4 
332.2 

72.7 
73.3 
74.0 
74.6 
75.2 

85.1 

69.0 

341 

9.3 

49.3 

75.9 

6 

8 
9 
10 

38  31.8o,, 
39  56  7J 
41  Sl^gH 
42  45.8(o/o 
44  lO.lj84'3 

1750.1LS1 
18  '58.2  £H 
20    5.3-'-' 
21  11.5  Xs 
22  16.7(65'2 

43  54.7 

J44  27.4 
44  58.8 
45  28.7 
45  57.3 

32.7 
31.4 
29.9 
28.6 

4951.7 

4941.0 
49  28.8 
49  15.1 
49    0.2 

10.7 

12.2 
13.7 
14.9 

34  30.2 
33  39.7 
3248.1 
31  55.4 
31  1.6 

50.5 
51.6 
52.7 
53.8 

2  16.3 
0  59.9 
59  43.0 
58  25.6 
57    7.6 

76.4 
76.9 
77.4 
78.0 

83.9 

!64.3 

272 

167 

54.9 

78.4 

11 
12 
13 
14 
15 

45  34.0  QO  7 
46  57.7  oo  ^ 
48  21.1  83'4 
4944.183'° 
51    6.7|82'6 

24  24.2  S'2 
25  26.4  S*'| 

26  27.6  "*•;: 

2727.660'0 

46  24.5 
46  50.2 

47  14.5 
47  37.4 
47  58.8 

25.7 
24.3 
22.9 
21.4 

48  43.5 
48  25.6 
48    6.3 
4745.5 
47  23.3 

17.9 
19.3 
20.8 

22.2 

30  6.7 

29  10.7 
28  13.7 
27  15.7 
26  16.6 

56.0 
57.0 
58.0 
59.1 

55  49.2 
54  30.3 
53  11.0 
51  51.3 
50  31.2 

78.9 
79.3 
79.7 
80.1 

182.2 

59.0 

200 

235 

60.0 

80.5 

16 
17 
18 
19 
20 

52  28.9  'ft1  o 
53  50.7  2}  « 

55  12.0  oi'3 
56  32.9  1™'  - 
5753.21 

28  26.6  _ft 
29  24.6  £*•" 
30  21.4  £!?•* 
31  17.0  r?'° 
3211.5!54'5 

48  18.8 
48  37.4 
48  54.5 
49  10.1 
49  24.4 

18.6 
17.1 
15.6 
14.3 

46  59.8 
46  34.8 
46    8.5 
45  40.7 
45  11.6 

25.0 
26.3 
27.8 
29.1 

25  16.6 
24  15.6 
23  13.6 
22  10.7 
21  6.8 

61.0 
62.0 
62.9 
63.9 

49  10.7 
47  49.9 
46  28.8 
45    7.5 
43  45.8 

80.8 
81.1 
81.3 
81.7 

79.8 

153.3 

127 

SO  4 

64.7 

81.9 

21 
22 
23 
24 
25 

59_13:0793 
0  32.3  78.7 
1  51.0|78  ! 

3    Q-V^ 
4  26.5| 

33    4.8' 
33  57.0  f  I 

34  47'9  49  8 
35  37.7,  TX^ 

36  26.2| 

4937.1 
49  48.3 
4958.1 
50    6.4 
50  13.3 

11.2 
9.8 
8.3 
6.9 

4441.2 
44    9.5 
43  36.4 
43    1.9 
42  26.2 

31.7 
33.1 
34.5 
35.7 

20  2.1 
18  56.4 
1749.9 
16  42.6 
15  34.4 

R,  „  42  23.9 
fifi  r   41     1.8 
°!^  39  39.5 
09  38  17.0 
a"*  3654.4 

82.1 
82.3 
82.5 
82.6 

76.8 

47.2 

54 

370 

8.9 

32,7 

26 
27 
28 
29 
30 

543.3 
659.4 
8  14.9 
929.6 
10  43.5 

76.1 
75.5 
74.7 
73.9 

37  13.4 
37  59.4 
3844.2 
39  27.6 
40    9.7 

46.0 
44.8 
43.4 
42.1 

50  18.7 
50  22.6 
50  25.0 
50  26.0 
50  25.5 

3.9 
2.4 
1.0 
0.5 

41  49.2 
41  10.8 
4031.2 
39  50.4 
39    8.3 

38.4 
39.6 
40.8 
42.1 

425.5 
3  15.7 
2  5.2 
10  54.0 
942.0 

9.8 
70.5 
71.2 
72.0 

3531.7 
34    8.8 
32  45.9 
31  23.0 
30    0.0 

82.9 
82.9 
82.9 
83.0 

go 

2° 

2° 

2' 

JO 

1° 

_J 

TABLE   L. 

Evection. 

Argument.     Evection,  corrected. 


73 


vn 

1  VII* 

VIII* 

IX* 

Xs 

XI« 

i 

i        i      i 

?  1°     Diff.v 

Diff.  0°           Diff.  0°           Diff.  0^ 

Diff.  0° 

Did 

u|X 

] 

/    „ 

'    "              '  " 

1    "             \   '    " 

,    „ 

tf 

030    0.0 

1  2337.0 

Q'  n  50  18.0 
BOO  49    6.0 

2051.7 
if"  20    9.6  ;„„ 

934.5   "       1950.3,,,. 
934.0    y-J  2032.4!7f. 

49  16.5 
50  30.4 

227  14.1 
325  51.2 
4  24  23.3 
523    5.6 

So  9  47  54.8 
o.7'q  46  44.3 
5H  4534.5 

82'7  44  25.6 

71.2 
70.5 
69.8 
68.9 

1849.2 
18  10.8 
17  33.8 

9  35.0    »•; 
937.4    J'J 
941.3    6* 
946.7    5'4 

O1     IRQ  ^"-^ 

22    oIK8 

22    ^46.0 
22  46.6|A7  9 

23  33.8  *7"* 

5145.1 
53    0.6 
5416.7 
55  33.5 

74.7 
76^8 

82.6 

682 

35.7 

6.9 

48.5 

77.4 

6J21  43.0 

?'20  20.5 
8|1S  53.2 
9J1736.1 
10|16  14.2 

82.5 
82.3 
82.1 
81.9 

43  17.4 
42  10.1 
41  3.6 
39  57.9 
38  53  2 

67.3 
66.5 
65.7 
64.7 

1658.1 
16  23.6 
15  50.5 
15  18.8 
14  48.4 

34.5 
33.1 
31.7 
30.4 

953.6    ... 

10              1.9            °g 

10  22:9  j1-2 

10  35.6,  12'7 

2422.3 
Ml«.l|S« 

26    3  0  ^'^ 
26  55.2  °2  I 
27  48.5  533 

56  50.9 
58    9.0 
59  27.7 
047.0 
2    6.8 

78.1 
78.7 
79.3 
79.8 

81.7 

63.9 

29.1 

14.3 

54.5 

80.3 

11  14  52.5 
12  1331.2 
13  12  JO  1 
14'l049.3 
15    9  28.8 

81.3 

81.1 
80.8 
80.5 

37  49.3 
36  46.4 
35  44.4 
34  43.4 
33  43.4 

62.9 
62.0 
61.0 
60.0 

14  19.3 
13  51.5 
13  25.2 
13    0.2 
1236.7 

27.8 
26.3 
25.0 
23.5 

1049.9,,, 

Ue   c'  AO.D 
1  T7  1 

11226  186 

114L220C 
12    1.2  20 

28  43.0 
29  38.6 
30  35.4 
31  334 
32  32.4 

55.6 

56.8 
58.0 
59.0 

3  27.1 
448.0!°°-; 
6    9.3  g}^ 
731.1!°^ 
8  53.3  82 

80.1 

59.1 

22.2 

21.4 

60.0 

82.6 

16    8    8.7 
17    649.0 
18    529.7 
19    4  10.8 
20    2  52.4 

79.7 
79.3 
78.9 
78.4 

32  44.3 
31  46.3 
30  49.3 
29  53.3 
28  58.4 

58.0 
57.0 
56.0 
54.9 

12  14.5 
11  53.7 
11  34.4 
11  16.5 

10  59.8 

20.8 
19.3 
17.9 
16.7 

1222.6 

18455  243 

13    9.8  24'3 

13  35.5  2_-7 

14    2.7  27'3 

33  32.4 
34  33.6 
35  35.8 
36  39.0 
37  43.3 

61.2 
62.2 
63.2 
64.3 

10  15.9       n 
1138.9^" 
13    8.3gJ; 

1426-°839 
15  49.9  8- 

78.0 

53.8 

14.9 

28.6 

65.2 

84.3 

21     134.4774 

22    017-0  7g  g 
23  59    0.1  76'4 
245743.775Q 
255627.Si 

28  4.6 
27  11.9 
26  20.3 
25  29.8 
24  40.5 

52.7 
51.6 
50.5 
49.3 

10  44.9 
1031.2 
10  19.0 
10    8.3 
959.0 

13.7 
12.2 
10.7 
9.3 

1431.3 
15    1.2 
15  32.6 
16    5.3 
16  39.4 

29.9 
31.4 
32.7 
34.1 

3848.5 
39  54.7 
41     1.8 
42    9.9 
43  18.9 

66.2 
67.1 
68.1 
69.0 

17  14.2  Q  ,  . 
18  38.6  Q*-* 
20    3.3  «47 

21  28.2  rr  , 

22  53.3  851 

175.2 

48.1 

7.8 

35.5 

69.8 

85.1 

&  56  12.6' 

27  53  53.0  JJJ 
28  52  44.0  '  *'" 
29  51  30.7  1?  * 
30  50  18.0 

23  52.4 
23  5.4 
22  19.6 
21  35.1 
2051.7 

47.0 
45.8 
44.5 
43.4 

951.2 
944.8 
939.9 
936.5 
934.5 

6.4 
4.9 
3.4 
2.0 

17  14.9 
1751.7 
18  29.9 
19    9.4 
19  50.3 

36.8 
38.2 
39.5 
40.9 

4428.7 
45  39.4 
46  51.0 
48    3.3 
49  16.5 

70.7 
71.6 
72.3 
73.2 

24  18.4 
25  43.7 
27    9.1 
28  34.5 
30    0.0 

85.3 
85.4 
85.4 
85.5 

lo°      i      |o° 

0° 

0° 

0° 

1° 

74 


TABLE   LI. 

Equation  of  Moon's  Centre. 
Argument.     Anomaly   corrected. 


0« 

I* 

II* 

III* 

IV* 

Yi 

7° 

Diff 
forlO 

10° 

Diff 
for  10 

12° 

Diff 
forlO 

13° 

Diff 
for  10 

12° 

Diff 
forlO 

9° 

Diff 
for  10 

O    ' 

f    // 

f    // 

/    /» 

/    // 

/    tt 

/•    ,/ 

0  0 
30 
1  0 
30 
2  0 
30 

0   0.0 

332.6 
7   5.2 
1037.8 
1410.3 

1742.7 

70.9 
709 
70.9 
70.8 
70.8 

2057.9 
2355.6 
26  52.2 
2947.7 
32  42.0 
3535.2 

59.2 
58.9 
58.5 
58.1 
57.7 

3843.6 
40  14.0 
41  42.7 
43    9.6 
4434.9 
4558.4 

30.1 
29.6 
29.0 

28.4 
27.8 

1735.2 
17209 
17   4.8 
1647.1 
1627.6 
16   6.5 

4.8 
5.4 
5.9 
6.5 
7.0 

1620.8 
1435.3 
1248.5 
11    0.4 
911.1 
720.5 

35.2 
35.6 
36.0 
36.4 
36.9 

5828.9 
5543.8 
5258.0 
5011.6 
4724.5 
4436.8 

55.0 
55.3 
55.5 
55.7 
55.9 

70.8 

57.3 

273 

7.6 

37.3 

56.1 

3  0 

30 
4  0 
30 
5  0 

21  15.0 
24  47.3 
28  19.4 
3151.2 
3523.0 

70.8 
70.7 
70.6 
70.6 

3827.1 
41  18.0 
44   7.6 
4656.0 
49  43.2 

57.0 
56.5 
56.1 
55.7 

4720.2 
48  40.3 
4958.7 
51  15.3 
5230.2 

26.7 
26.1 
25.5 
25.0 

1543.7 
1519.2 
1453.1 

1425.2 
1355.8 

8.2 
8.7 
9.3 
9.8 

528.7 
335.6 
141.3 
5945.8 
5749.1 

37.7 
38.1 
38.5 
38.9 

41  48.5 
38  59.5 
3610.0 
3319.8 
3029.1 

56.3 
56.5 
56.7 
56.9 

70.5 

55.3 

24.4 

10.4 

39.3 

57.1 

30 
6  0 

30 
7  0 
30 

3854.5 
42  25.8 
45  56.9 
4927.7 
52  58.2 

70.4 
70.4 
70.3 
70.2 

5229.1 
55  13.8 
5757.2 

54.9 
54.5 
54.0 
53.6 

5343.3 
5454.7 
56    4.4 
57  12.3 

5818.5 

23.8 
23.2 
22.6 
22.1 

1324.7 
1251.9 
1217.4 
1141.4 
11    3.7 

10.9 
11.5 
12.0 
12.6 

5551.1 
5352.0 
5151.7 
49  50.3 
4747.6 

39.7 
40.1 
40.5 
40.9 

2737.8 
2445.9 
2153.5 
19    0.6 
16    7.1 

57.3 
57.5 
57.0 

57.8 

039.3 
320  1 

70.1 

53.2 

21.5 

13.1 

41.3 

58.0 

8  0 
30 
9  0 
30 
10  0 

56  28.5 
59  58.4 

70.0 
69.9 
69.7 
69.6 

559.7 
837.9 
11  14.8 
1350.3 
1624.5 

52.7 
52.3 
51.8 
51.4 

5922.9 
~025^ 
126.5 
225.7 
323.0 

20.9 
20.3 
19.7 
19.1 

1024.3 
943.4 
9    0.8 
816.6 
730.8 

13.6 
14.2 
14.7 
15.3 

4543.8 
4338.9 
4132.8 
39  25.6 
37  17.3 

41.7 
42.0 

42.4 
42.8 

1313.1 
1018.6 
723.6 
428.1 
1  32.2 

58.2 
58.3 
58.5 
58.6 

328.0 
657.2 
1026.0 

695 

50.9 

18.fi 

15.8 

43,1 

58.8 

30 
11  0 
30 
12  0 
30 

1354.5 
1722.5 
2050.1 
2417.3 
2744.0 

69.3 
69.2 
69.1 
68.9 

1857.3 
21  28.8 
2358.8 
2627.5 
28  54.7 

50.5 
50.0 
49.6 
49.1 

418.7 
512.5 
6   4.6 
654.9 
743.5 

17.9 
17.4 
16.8 
16.2 

643.4 
554.4 
5   3.9 
411.7 
318.0 

16.3 
16.8 
17.4 
17.9 

35    7.9 

3257.4 
3045.8 
2833.1 
26  19.4 

43.5 
43.9 
44.2 
44.6 

5S35.8 
5538.9 
5241.7 
4943.9 
4645.8 

59.0 
59.1 
59.3 
59.4 

68.7 

48.6 

15.6 

18.4 

44.9 

59.5 

13  0 
30 
14  0 
30 
15  0 

31  10.2 
34  35.8 
38    1.0 
41  25.6 
4449.6 

68.5 
68.4 
68.2 
68.0 

31  20.5 
3344.9 
36    7.9 
3829.4 
4049.3 

48.1 
47.7 
47.2 
46.6 

830.3 
915.4 
958.6 
1040.1 
11  19.9 

15.0 
14.4 
13.8 
13.3 

222.7 
125.8 
027.4 
59  27.4 
58  25.9 

19.0 
19.5 
20.0 
20.5 

24   4.6 
2148.8 
1931.9 
1714.1 
1455.2 

45.3 
45.6 
45.9 
46.3 

4347.3 
4048.4 
3749.1 
3449.5 
31494 

59.6 
59.8 
59.9 
60.0 

8° 

11° 

13° 

12° 

11° 

8° 

TABLE  LI. 

Equation  of  Moon's  Centre. 
Argument.      Anomaly  corrected. 


75 


VI* 

[  VII* 

VIII* 

IX* 

X* 

Xf« 

DiffLrt 

Diff 

Diff 

Diff  ,0 

Diff 

30 

jh-r, 

7° 

forlO 

4 

forlO 

1 

forlO 

0 

for  10  l 

for  10 

fo  10 

O      ' 

/    // 

/     // 

,    „ 

/    // 

/    // 

/    // 

0  0 
30 
1  0 
30 
2  0 
30 

0   0.0  R[  g 

56546;61  8 
5349.2  6!  8 
50  43.9  61  8 

4738.66L7 
4433.4 

131.1 
5846.7 
56   3.0 
5320.0 
50  37.7 
4756.2 

54.8 
54.6 
543 
54.1 
53.8 

43  39.2 
41  55.0 
40  12.0 
3830.5 
36  50.3 
3511.3 

34.7 
34.3 
33.8 
KA 
33.0 

42  24.8 
4212.1 
42  1.2 
41  52.0 
4144.4 
41  38.7 

4.2 
3.6 
3.1 
2.5 
1.9 

21  16.4 
22  48.5 
2422.2 
25  57.7 
2734.8 
29  13.7 

30.7 
31.2 
31.8 
32.4 
33.0 

39    21 
43    O.b 
41    f.V 
4*    1.7 
51    3.7 
54    6.7 

L9.6 
60.0 
60.3 
60.7 
61.0 

61.8 

53.6 

32.5 

1.4 

33.5 

61.3 

3  0 
30 
4  0 
30 
5  0 

4128.1 
38  23.0 
35  18.0 
32  13.0 
29   8.1 

61.7 
61.7 
61.7 
61.6 

45  15.4 
4235.3 
3956.0 
37  17.4 
3439.6 

53.4 
53.1 
52.9 
52.6 

3333.7 
31  57.5 
30  22.6 
2849.0 
2716.8 

32.1 
31.6 
31.2 
30.7 

41  34.6 
41  32.2 
4131.6 
41  32.7 
41  35.6 

0.8 
0.2 
0.4 
1.0 

30  54.2 
3236.3 
34  20.2 
36    5.6 
3752.8 

34.0 
34.6 
35.1 
35.7 

57    0.7 
0/5.8 
321.8 
628.8 
936.8 

61.7 
62.0 
62.3 
62.7 

61.6 

52.3 

30.2 

1.5 

36.2 

63.0 

30 
6  0 
30 
7  0 
30 

26   3.4 

22  58.8 
1954.3 
1650.0 
1345.8 

61.5 
61.5 
61.4 
61.4 

32   2.7 

2926.5 
2651.1 
24  16.6 
2142.9 

52.1 
51.8 
51.5 
51.2 

2546.1 
2416.7 
2248.7 
2122.1 
1956.9 

29.8 
29.3 
28.9 
28.4 

4140.1 
41  46.4 
41  54.5 
42  4.3 
42  15.9 

2.1 
2.7 
3.3 
3.9 

3941.5 
41  32.0 
43  24.0 
4517.7 
47  12.9 

36.8 
37.3 
37.9 

38.4 

1245.7 
1555.5 
19    6.2 
22  17.8 
2530.3 

63.3 
63.6 
63.9 
64.2 

61.3 

51.0 

27.9 

4.4 

39.0 

64.5 

8  0 
30 
9  0 
30 
10  0 

1041.9 

738.0 
434.4 
131.0 
58~27^8 

61.3 
61.2 
61.1 
61.1 

1910.0 
1633.0 
14   6.9 
1136.6 
9    7.3 

50.7 
50.4 
50.1 
49.8 

1833.1 
1710.8 
1549.8 
1430.4 
1312.5 

27.4 
27.0 
265 
26.0 

4229.2 
42  44.2 
43  1.1 
4319.6 
43  39.9 

5.0 
5.6 
6.2 
6.8 

49    9.8 
51    8.3 
53   8.4 
5510.1 
57  13.3 

39.5 
40.0 
40.6 
41.1 

2843.7 
3157.8 
35  12.9 
3828.7 
41  45.2 

64.7 
65.0 
65.3 
65.5 

61.0 

49.5 

25.5 

74 

41.6 

65.8 

30  55  24.  9  ft, 

11   Oj  52  22.2  ° 
304919.7^ 
1204617.5^7 

30  43  15.6  60'6 
160.5 

638.9 
411.3 
144.7 

49.2 
48.9 
48.6 
48.2 

47.9 

1155.9 
1040.9 
927.3 
815.2 
7   4.6 

25.0 
24.5 
24.0 
23.5 
23.1 

44  2.0 

4425.9 
4451.5 
45  18.8 
4548.0 

8.0 
8.5 
9.1 
9.7 
10.3 

59  18.2 
124T5 
332.4 
541.9 
752.9 

42.1 
42.6 
43.2 
43.7 

44.2 

45   2.6 
48  20.7 
5139.6 
5459.1 
5819.3 

66.0 
66.3 
66.5 
66.7 

67.0 

59  18.9 
56  54.2 

13  0 
30 
14  0 
30 
15  0 

40  14.0 
3712.6 
3411.6 
31  10.9 
28  10.6 

60.5 
60.3 
60.2 
60.1 

54  30.4 
52   7.5 
4945.6 
4724.7 
45   4.8 

47.6 
47.3 
47.0 
46.6 

555.4 
447.8 
341.7 
237.1 
134.1 

22.5 
22.0 
21.5 
21.0 

46  18.9 
4651.5 
4726.0 
48  2.2 
4840.1 

10.9 
11.5 
12.1 
12.6 

10   5.5 

1219.5 
1435.1 
1652.1 
1910.7 

44.7 
45.2 
45.7 
46.2 

140.3 
5    1.9 
824.1 
1146.9 
15104 

67.2 
67.4 
67.6 
67.8 

5° 

2° 

1° 

0° 

2° 

5° 

76 


TABLE   Ll. 

Equation  of  Moon's  Centre 
Argument.     Anomaly  corrected. 


r* 

0* 

I* 

II* 

III* 

IV* 

V* 

8° 

Diff 
for  10 

11° 

M'     o 
forlO  13 

Diff 
forlO 

12° 

Diffi.o 
for  10,  H 

Diff 
forlO 

8° 

Diff 
for  10 

O    ' 

/    // 

/        V 

rr    \ 

/    /> 

,     // 

w 

,    ., 

,    „ 

15  0 

30 
16  0 
30 
17  0 
30 

4449.6 
4813.1 
5135.9 
5458.1 
58197 

67.8 
67.6 
67.4 
67.2 
67.0 

4049.3 
43    7.9 
4524.9 
4740.5 
49  54.5 
52    7.1 

46.2 
45.7 

45.2 
44.7 
44.2 

11  19.9 
1157.8 
1234.0 
13    8.5 
1341.1 
1412.0 

12.6 
12.1 
11.5 
10.9 
10.3 

5825.9 
5722.9 
56  18.3 
5512.2 
54  4.6 
5255.4 

21.0 
21.5 
22.0 
22.5 
23.1 

1455.2 
1235.3 
1014.4 
752.5 
529.6 
3    5.8 

46.6 
47.0 
47.3 
47.7 
47.9 

31  49.4 
2849.1 
25  48.4 
22  47.4 
1946.0 
1644.4 

no.i 

60.2 
60.3 
60.5 
60.5 

140  7 

66.7 

43.7 

9.7 

23.5 

483 

60.6 

18  0 
30 
19  0 
30 
20  0 

5    0.9 

820.4 
1139.3 
1457.4 
18  14.8 

66.5 
66.3 
66.0 
65.8 

5418.1 
5627.6 
5835.5 

43.2 
42.6 
42.1 
41.6 

1441.2 
15    8.5 
1534.1 
1558.0 
1620.1 

9.1 

8.5 
8.0 
7.4 

51  44.8 
5032.7 
4919.1 
48  4.1 
4647.5 

24.') 
24.5 
25.0 
25.5 

041.1 
58T5T3 
5548.7 
5321.1 
5052.7 

48.6 
48.9 
49.2 
49.5 

1342.5 
1040.3 
737.8 
435.1 
132.2 

60.7 
60.8 
60.9 
61.0 

041.8 
246  7 

65.5 

41.1 

6.8 

26.0  | 

49.8 

61.1 

30 

21  0 
30 
22  0 
30 

2131.3 
2447.1 
28    2.2 
3116.3 
3429.7 

65.3 
65.0 
64.7 
64.5 

449.9 
651.6 
851.7 
1050.2 
1247.1 

40.6 
40.0 
39.5 
39.0 

1640.4 
1658.9 
1715.8 
1730.8 
1744.1 

6.2 
5.6 
5.0 

44 

4529.6 
4410.2 
42  49.2 
4126.9 
40  3.1 

26.5 
27.0 

27.4 
27.9 

48  23.4 
4553.1 
4322.0 
40  50.0 
38  17.1 

50.1 
50.4 
50.7 
51.0 

5829.0 
5525.6 
5222.0 
4918.1 
46  14.2 

61.1 
61.2 
61.3 
61.3 

64.2 

38.4 

3.9 

28.4 

51.2 

61.4 

23  0 
30 
24  0 
30 
25  0 

3742.2 
40  53.8 
44   4.5 
47  14.3 
50  23.2 

63.9 
63.6 
63.2 
63.0 

1442.3 
1636.0 
18  28.0 
20  18.5 
22    7.2 

37.9 
37.3 
36.8 
36.2 

1755.7 
18    5.5 
1813.6 
1819.9 
1824.4 

33 

2.7 
2.1 
1.5 

38  37.9 
3711.3 
35  43.3 
34  13.9 
32  43.2 

28.9 
29.3 
29.8 
30.2 

3543.4 
33   8.9 
3033.5 
2757.3 
2520.4 

51.5 
51.8 
52.1 
52.3 

43  10.0 
40    5.7 
37    1.2 
3356.6 
3051.9 

61.4 
61.5 
61.5 
61.6 

62.7 

35.7 

1  0 

307 

52fi 

61  6 

30 
26  0 
30 
27  0 
30 

5331.2 
5639.2 
5944.2 

62.3 
62.0 
61.7 
61.3 

23  54.4 

2539.8 
2723.7 
29    5.8 
30  46.3 

35.1 
34.6 
34.1 
33.5 

1827.3 

1828.4 
1827.8 
1825.4 
1821.3 

0.4 
0.2 
0.8 
1.4 

31  11.0 
2937.4 
28  2.5 
2626.3 

2448.7 

31.2 
31.6 
32.1 
32.5 

2242.6 
20    4.0 
1724.7 
1444.7 
12    3.8 

52.9 
53.1 

3.3 
53.6 

2747.0 
2442.0 
21  37.0 
1831.8 
1526.6 

61.7 
61.7 
61.7 
61.7 

249.3 
5  53.3 

61.0 

33.0 

1.9 

330 

538 

61  7 

28  0 
30 
29  0 
30 
30  0 

856.3 
1158.3 
1459.3 
1759.2 
2057.9 

60.7 
60.3 
60.0 
59.6 

3225.2 
34   2.3 
35  37.8 
3711.5 
38  43.6 

32.4 
31.8 
31.2 
30.7 

1815.6 
18    8.0 
1758.8 
1747.9 
1735.2 

2.5 
3.1 
3.6 

4.2 

23  9.7 
21  29.5 
1948.0 
18  5.0 
1620.8 

33.4 
33.8 
34.3 
34.7 

922.3 
640.0 
357.0 
1  133 

54.1 
54.3 
54.6 

54.8 

1221.4 
916.1 
610.8 
3    54 
0    0.0 

61.8 
61.8 
61.8 
61.8 

58  28.9 

10° 

|l2° 

13° 

12° 

9° 

7° 

TABLE  LI 


77 


Equation  of  Moon 's  Centre. 
Argument.      Anomaly  corrected. 


VI* 

VII* 

VIII* 

IX* 

X* 

XL 

1 

5° 

Diff 
for  10 

2° 

Diff  '!  0 
forW1 

Diff 
for  10 

0° 

Diff20 

for  10  2 

Diff 
for  10 

5° 

Diff, 

for  Id 

O      ' 

,    „ 

/    „ 

,    „ 

.    „ 

.    ,. 

/    /. 

15  0  28  10.6    " 
302510.5™° 
16  022  10.9.  5?Q-r 

30  wn.6  ™ 

17  016  M.7  JJ-J 

30  13  14.2  * 

45   4.8 
42  45.9 
4028.1 
3811.2 
35  55.4 
3340.6 

46.3 
45.9 
45.6 
45.3 
44.9 

134.1 
032.6 

20.5 
20.0 
19.5 
19.0 
18.4 

4840.1 
4919.9 
50    1.4 
5044.6 
5129.7 
52  16.5 

13.3 

13.8 
14.4 
15.0 
15.6 

1910.7 
21  30.6 
2352.1 
2615.1 
2839.5 
31    5.3 

46.6 
47.2 
47.7 
48.1 
48.6 

151014 
834.4 
2159.0 
25  24.2 
28  49.8 
32  16.0 

68.0 
68.2 
68.4 
68.5 
68.7 

5932.6 
.58  34.2 
5737.3 
56  42  0 

1 

59.4 

44.6 

17.9 

16.2 

49.1 

68.9 

18  0 
30 
19  0 
30 
20  0 

1016.1 
718.3 
421.1 
124.2 

59.3 
59.1 
59.0 
58.8 

31  26.9 

2914.2 
27    2.6 
2452.1 
2242.7 

44.2 
43.9 
43.5 
43.1 

5548.3 
5456.1 
54   5.6 
5316.6 
5229.2 

17.4 
16.8 
16.3 
15.8 

53    5.1 
5355.4 
5447.5 
5541.3 
56  37.0 

16.8 
17.4 
17.9 
18.6 

3332.5 
36    1.2 
3831.2 
41    2.7 
43  35.5 

49.6 
50.0 
50.5 
50.9 

3542.7 
39    9.9 
42  37.5 
46   5.5 
49  34.0 

69.1 
69.2 
69.3 
69.5 

58  27.8 

58.6 

42.8 

15.3 

19.1 

51.4 

69.6 

305531.9 
21  05236.4 
304941.4 
22  04646.9 
30  43  52.9 

58.5 
58.3 
58.2 
58.0 

20  34.4 
1827.2 
1621.1 
1416.2 
1212.4 

42.4 
42.0 
41.6 
41.3 

51  43.4 
50  59.2 
5016.6 
4935.7 
48  56.3 

14.7 
14.2 
13.6 
13.1 

5734.3 
58  33.5 
59  34.4 
"037.] 
1  41  5 

19.7 
20.3 
20.9 
21.5 

46    9.7 
48  45.2 
5122.1 
54    0.3 
56  39.9 

51.8 
52.3 
52.7 
53.2 

53    2.8 
5631.9 

69.7 
69.9 
70.0 
70.1 

0    1.6 
331.5 

718 

57,8 

40.9 

12.6 

22.1 

536 

70.2 

23  04059.4 
30  38    6.5 
24  03514.1 
303222.2 
25  0,2930.9 

57.6 
57.5 
57.3 
57.1 

10    9.7 
8   8.3 
6    8.0 
4   8.9 
210.9 

40.5 
40.1 
39.7 
39.3 

4818.6 
4742.6 
47   8.1 
46  35.3 
46   4.2 

12.0 
11.5 
10.9 
10.4 

247.7 
3  55.6 
5   5.3 
616.7 
729.8 

22.6 
23.2 

23.8 
24.4 

5920.7 

54.0 
54.5 
54.9 
55.3 

1032.3 
14   3.1 
1734.2 
21    5.5 
2437.0 

70.3 
70.4 
70.4 
70.5 

2   2.8 
446.2 
730.9 
1016.8 

56.9 

38.9 

9.8 

25.01 

55.7 

70.6 

302640.2  ,ft7 

26  02350.0  ?;« 

3021    0.5^ 
27  MieiUMjJ-J 

30  1523.2  5b'X 

014.2 
58  18.7 
50  24.4 
J431.3 
5239.5 

38.5 
38.1 
37.7 
37.3 

45  34.8 
45    6.9 
4440.8 
4416.3 
43  53.5 

9.3 
8.7 
8.2 
7.6 

844.7 
10    1.3 
11  19.7 
1239.8 
14    1.6 

.,_  c   13    4.0'   ,.  _ 

2o"5  15524561 
26.1   Jo42o56.5 

26.7  2^32  o  57.0 
27.3  »*jki 

28    8.S 
3140.7 
35  12.8 
3845.1 
4217.3 

70.6 
70.7 
70.7 

70.8 

559 

36.9 

7.0 

27.8                57.7 

70.8 

28  0 
30 
29  0 
30 
30  0 

1235.5 
948.41 
7   2.0 
416.2 
131.1 

55.7 
55.5 
55.3 
55.0 

5048.9 
48  59.6 
4711.5 
4524.7 
4339.2 

oft  ,  4332.4 
,"•*  4312.9 
3?'°  42  55.2 

352  4239'1 
d5^  4224.8 

6.5 
5.9 

5.4 
4.8 

1525.1 
1650.41 
1817.3 
1946.0 
21  16.4 

2718.0 

60.1:     OA   1f>   0    «W,1 

29.0  •£  12  \  58.5 
29.6  ™   I  J  53.9 
30.1  5   I'*  69.2 

4549.7 
49  22.2 
5254.8 
50  27.4 
0   OT6 

70.8 
70.9 
70.9 
70.9 

4° 

!i° 

0° 

1° 

3° 

7° 

\ 

78 


TABLE   LII. 
Variation. 

Argument.     Variation,  corrected. 


0* 

I* 

II* 

Ills 

IV* 

v« 

?o° 

DiffJl0 

Diff. 

1° 

Diff. 

Oo 

Diff. 

0° 

Diff. 

0° 

Diff. 

K)   | 

0 

I 

2 
3 
4 
ft 

38    0.0 
3^13.3 
40  26.5 
41  39.5 
42  52.2 
44    4.5 

73.3 
73.3 
73.0 

72.7 
72.3 

8    1.5 
835.5 
9    7.2 
9  36.5 
10    3.4 
10  27.9 

34.0 
31.7 
29.3 
26.9 
24.5 

657.9 
6  18.0 
5  35.9 
451.7 
4    5.5 
3  17.3 

39.9 
42.1 
44.2 
46.2 
48.2 

35  54.4 
34  40.4 
33  26.6 
32  13.0 
30  59.6 
29  46.7 

74.0 
73.8 
73.6 
73.4 
72.9 

529.5 
454.2 
421.3 
350.6 
322.3 
256.5 

35.3 
32.9 
30.7 

28.3 
25.8 

6  1.6 

641.6 
723.9 
8  8.4 
8  55.0 
943.7 

40.0 
42.3 

44.5 
46.6 

48.7 

71.9 

22.0 

50.1 

72.4! 

23.4 

50.8 

6 
7 
8 
9 
10 

45  16.4 
46  27.7 
47  38.4 
48  48.3 
49  57.4 

71.3 
70.7 
69.9 
69.1 

10  49.9 
11    9.4 
11  26.4 
11  40.9 
11  52.9 

19.5 
17.0 
14.5 
12.0 

227.2 
1  35.3 
041.6 
5946.1 
5849  0 

51  9  28  34'3 

53  7  27  22'4 
£?•'  126  11.2 

^-?i25    0.7 
°''1   2351.1 

71.9 

71.2 
70.5 
69.6 

233.1 

2  12.1 
1  53.7 
1  37.8 
124.5 

21.0 
18.4 
15.9 
13.3 

10  34.5 
11  27.3 
12  22.0 
13  18.6 
14  16.9 

52.8 
54.7 
56.6 
58.3 

68.2 

9.3 

58.8 

68.8 

10.8 

60.1 

11 
12 
13 
14 
15 

51    5.6 
52  12.8 
53  18.9 
54  23.8 
55  27.5 

67.2 
66.1 
64.9 
63.7 

12    2.2 
12    9.0 
12  13.2 
12  14.8 
12  13.9 

6.8 
4.2 
1.6 
0.9 

57  50.2 
56  50.0 
55  48.3 
54  45.2 
53  40.9 

60.2 
61.7 
63.1 
64.3 

22  42.3 
21  34.5 
20  27.9 
19  22.3 
18  18.0 

67.8 
66.6 
65.6 
64.3 

1  13.7 
1    5.5 
1    0.0 
057.0 
056.7 

8.2 
5.5 
3.0 
0.3 

15  17.0 
16  18.7 
17  22.0 
18  26.9 
19  33.1 

61.7 
63.3 
64.9 
66.2 

62.3 

3.6 

65.6 

63.0 

2.3 

67.6 

16 
17 

18 
19 
20 

56  29.8 
57  30.7 
5830.1 
59  28.0 

60.9 
59.4 
57.9 
56.2 

12  10.3 
12    4.2 
11  55.5 
11  44.2 
11  30.5 

6.1 
8.7 
11.3 
13.7 

52  35.3 

51  28.5 
50  20.7 
49  11.9 
48    2.2 

66.8 
67.8 
68.8 
69.7 

17  15.0 
16  13.4 
15  13.2 
14  14.6 
13  17.5 

fii  fi  °  59-° 
61.6  L     qq 

60.2  i!  *; 

KQ  P.   1  11.5 
oo.  b   ,  91  R 
c7  i    A  *!.*» 
5711   134.4 

4.9 
7.6 
10.1 

12.8 

20  40.7 
21  49.6 
22  59.6 
24  10.8 
25  22.9 

68.9, 
70.0, 
71.2 
72.1 

0  24  2 

54.5 

16.4 

70.5 

55.3 

15.4 

73.0 

21 
22 
23 
24 
25 

1  18.7 
2  11.4 
3    2.3 
351.2 
438.2 

52.7 
50.9 
48.9 
47.0 

11  14.1 
10  55.3 
10  34.0 
10  10.2 
944.0 

18.8 
21.3 
23.8 
26.2 

4651.7L  „   1222.2 
45  40.5  1'!  Q   11  28.5 
44  28.6  ill*   1036.7 
4316.ll';-;!     946.8 
42    3.2i/x5'y     858.8 

53.7 
51.8 
49.9 
48.0 

149.8 
2    7.8 
2  28.3 
251.4 
3  16.9 

18.0 
20.5 
23.1 
25.5 

26  35.9 
27  49.8 
29  4.5 
30  19.7 
31  35.6 

73.9 
74.7 
75.2 
75.9 

44.9 

28.6                173.3 

46.1 

28.1 

76.3 

26 
27 

28 
29 
30 

523.1 
6    6.0 

646.7 
7  25.2 
8    1.5 

42.9 
40.7 
38.5 
36.3 

9  15.4 

8  44.5 
8  11.2 
735.7 
6  57.9 

30.9 
33.3 
35.5 

37.8 

4049.9!7,,7 
39  36.2^  ' 
38  22.4  JJJ 
a-/    a  A  74-° 
Ai    8-4740 
35  54.4  ^ 

8  12.7 
728.7 
646.8 
6    7.1 
529.5 

44.0 
41.9 
397 
37.6 

345.0 
4  15.6 
448.5 
523.9 
6    1.6 

30.6 
32.9 
35.4 
37.7 

3251.9 
34  8.6 
35  25.6 
36  42.7 
38  0.0 

76.7 
77.0 
77.1 
77.3 

1° 

1° 

|0° 

loo 

0° 

0° 

TABLE  LII. 
Variation. 

Argument.     Variation  corrected. 


79 


VI- 

VII* 

VIII* 

IX* 

X* 

XL 

i 

1 

0° 

Diff.  1° 

Diff. 

1° 

Diff.  Oo 

Diff. 

0° 

Diff. 

0° 

Diff. 

0 

1 

2 
3 

4 
5 

38    0.0 
39  17.3 
40  34.4 
41  51.4 
43    8.1 
4424.4 

77.3 
77.1 
77.0 
76.7 
76.3 

958.4 
1036.1 
11  11.5 
1144.4 
1215.0 
1243.1 

37.7 
35.4 
32.9 
30.6 
28.1 

1030.5 
952.9 
913.2 
831.3 
747.3 
7    1.2 

37.6 
39.7 
41.9 
44.0 
46.1 

40   5.6 
3851.6 
3737.6 
3623.8 
3510.1 
3356.8 

74.0 
74.0 
73.8 
73.7 
73.3 

9  2.1 

824.3 
748.8 
715.5 
644.6 
616.0 

37.8 
35.5 
33.3 
30.9 
28.6 

758.5 
834.8 
913.3 
954*0 
1036.9 
1121.8 

36.3 

38.5 
40.7 
429 
44.9 

75.9 

25.5 

48.0 

72.9 

26.2 

47.0 

6 

7 
S 
9 
10 

45  40.3 
46  55.5 
4810.2 
4924.1 
5037.1 

75.2 
74.7 
73.9 
73.0 

13  8.6 
1331.7 
1352.2 
1410.2 
1425.6 

23.1 
20.5 
18.0 
15.4 

613.2 
523.3 
431.5 
337.8 
242.5 

49.9 
51.8 
53.7 
55.3 

32  43.9 
3131.4 
3019.5 
29    8.3 
2757.8 

72.5 
71.9 
71.2 
70.5 

5  49.8 
526.0 
5  4.7 
445.9 
429.5 

23.8 
21.3 
18.8 
16.4 

12   8.8 
1257.7 
1348.6 
1441.3 
1535.8 

48.9 
50.9 
52.7 
54.5 

72.1 

12.8 

57.1 

69.7 

13.7 

56.2 

11 
12 
13 
14 
15 

5149.2 
53    0.4 
54  10.4 
5519.3 
5626.9 

71.2 
70.0 
68.9 
67.6 

1438.4 
1448.5 
1456.1 
15  1.0 
15  3.3 

10.1 
2.3 

145.4 
046.8 

58.6 
60.2 
61.6 
63.0 

2648.1 
25  39.3 
2431.5 
23  24.7 
2219.1 

68.8 
67.8 
66.8 
65.6 

4  15.8 
4  4.5 
355.8 
349.7 
346.1 

11.3 
8.7 
6.1 
3.6 

1632.0 
1729.9 
1829.3 
1930.2 
20  32.5 

57.9 
59.4  i 
60.9  ! 
62.3  ! 

5946.6 
58  45.0 
5742  0 

66.2 

0.3 

64.3 

64.3 

0.9 

63.7 

16 
17 
18 

19 
20 

5733.1 

5838.0 
5941.3 

64.9 
63.3 
61.7 
60.1 

15  3.0 
15  0.0 
1454.5 
1446.3 
1435.5 

3.0 
5.5 

8.2 
10.8 

56  37.7 
5532.1 
5425.5 
53  17.7 
52   8.9 

65.6 
66.6 
67.8 
68.8 

21  14.8 
2011.7 
1910.0 
18    9.8 
1711.0 

63.1 
61.7 
60.2 
58.8 

345.2 
346.8 
351.0 
357.8 
4  7.1 

1.6 
4.2 
6.8 
9.3 

21  36.2 
2241.1 
23  47.2 
2454.4 
26   2.6 

64.9 
66.1 
67.2 
68.2 

043.0 
1  43  1 

58.3 

13.3 

69.6 

57.1 

12.0 

69.1 

21 
22 
23 
24 
25 

241.4 
S38.0 
432.7 
525.5 
616.3 

56.6 
54.7 
153.8 

50.8 

1422.2 
14  6.3 
13  47.9 
1326.9 
13  3.5 

159  5059'3 
844948-8 
i,  J  !48  37.6 

Jj'J  4725.7 
^  *  46  13.3 

70.5 
71.2 
71.9 
72.4 

1613.9  ,,  - 
1518.4^ 

1424'7!519 

13328  50? 
1242.7|m 

419  1 
433.6 
450.6 
510.1 
532.1 

14.5 
17.0 
19.5 
22.0 

2711.7 
2821.6 
29  32.3 
3043.6 
31  55.5 

69.9 
70.7 
71.3 
71.9 

48.7 

25.8 

72.9 

48.2 

24.5 

72.3 

26 
27 
28 
29 
30 

7   5.0 
751.6 
836.1 
918.4 
958.4 

46.6 
44.5 
42.3 
40.0 

1237.7 
12  9.4 
1138.7 
11  5.8 
1030.5 

S'B^ 

jg-JUas3.4 

q^o  J41  19.6 
35'3  40    5.6 

73.4 
173.6 
73.8 
74.0 

1154.5 
11    8.3 
1024.1 
942.0 
9    2.1 

46.2 
44.2 
42.1 
39.9 

556.6 
623.5 
652.8 
724.5 
758.5 

26.9 

S? 

34.0 

33    7.8 
34  20.5 
35  33.5 
36  46.7 
38    0.0 

72.7 
73.0 
73.2 
73.3 

1° 

1° 

loo 

Oo 

0° 

0° 

TABLE  LTII.     Reduction. 
Argument.     Supplement  of  Node  +  Moon's  Orbit  Longitude. 


Oa  Vis 

Diff. 

Is  VIIs  Diff.   Us  VIIIs  Diff. 

His  IXs 

Diff 

IVsXs 

Diff. 

VsXI* 

Diff. 

0 

,      „ 

.     // 

/     „ 

/    „ 

/     „ 

,      „ 

0 

1 

2 
3 
4 
5 

7     0.0 
6  45.6 
6  31.2 
6  16.9 
6     2.6 
5  48.4 

14.4 
14.4 
14.3 
14.3 
14.2 

1     3.0  " 
0  56.0  J'J 
0  49.5  !'f 
0  43.4  H 
0  37.8  £, 
0  32.7  D1 

3.0 
10.4 
18.3 
26.5 
35.2 
44.2 

7.4 
7.9 
8.2 
8.7 
9.0 

7    0.0 

7  14.4 
7  28.8 
7  43.1 
7  57.4 
8  11.6 

14.4 
14.4 
14.3 
14.3 
14.2 

12  57.0 
13     4.0 
13   10.5 
13  16.6 
13  22.2 
13  27.3 

7.0 
6.5 
6.1 
5.6 
5.1 

12  57.0 
12  49.6 
12  41.7 
12  33.5 
12  24.8 
12   15.8 

7.4 
7.9 
8.8 
8.7] 
9.0J 

14.1 

4.5 

9.5 

14.1 

4.5 

9.5 

6 

7 
8 
9 
10 

5  34.3 
5  20.3 
5     6.4 
4  52.6 
4  39.0 

14.0 
13.9 
13.8 
13.6 

0  28.2 
0  23.9  JJ 
0  20.0  ;* 
0  16.8  *•* 

6  144  1 

1  53.7 
2     8.5 

2  13.7 
2  24.2 
2  35.0 

9.8 
10.2 
10.5 
10.8 

8  25.7 
8  39.7 
8  53.6 
9    7.4 
9  21.0 

14.0 
13.9 
13.8 
13.6 

13  31.8 
13  36.1 
13  40.0 
13  43.2 
13  45.9 

1 

12     6.3 
11   56.5 
11   46.3 
11   35.8 
11   25.0 

9.8 
10.2 
10.5 
10.8 

13.4 

2.3 

11.2 

13.4 

2.3 

(11.2 

11 
12 
13 
14 
15 

4  25.6 
4  12.3 
3  59.3 
3  46.5 
3  33.9 

13.3 
13.0 
12.8 
12.6 

0  11.8 
0  10.1  J'j 

J  ™™ 

2  46.2 
2  57.7 
3     9.5 
3  21.6 
3  33.9 

11.5 

11.8 
12.1 
12.3 

9  34.4 
9  47.7 
10    0.7 
10  13.5 
10  26.1 

13.3 
13.0 

12.8 
12.6 

13  48.2 
13  49.9 
13  51.2 
13  51.9 
13  52.2 

1.7 
1.3 
0.7 
0.3 

11    13.8 
11     2.3 
10  50.5 
10  38.4 
10  26.1 

11.5 
11.8 
12.1 
12.3 

12.3 

0.3 

12.6                 12.3 

0.3 

12.6 

16 
17 

18 
19 
20 

3  21.6 
3     9.5 

2  57.7 
2  46.2 
2  35.0 

12.1 
11.8 
11.5 
11.2 

0     8.1 
0    8.8 
0  10.1 
0  11.8 
0  14.1 

0.7 
1.3 
1.7 
2.3 

3  46.5 
3  59.3 
4  12.3 
4  25.6 
4  39.0 

12.8 
13.0 
13.3 
13.4 

10  38.4 
1050.5  ,}*•* 
11     2.3  .}}•?. 
1113.8       '* 
11  25.0  1L2 

13  51.9 
13  51.2 
13  49.9 
13  48.2 
13  45.9 

0.7 
1.3 
1.7 
2.3 

10   13.5 
10     0.7 

9  47.7 
9  34.4 
9  21.0 

12.8 
13.0 
13.3 
13.4 

10.8 

2.7 

13.6 

10.8J 

2.7 

13.6 

21 
22 
23 
24 
25 

2  24.2 
2  13.7 
2     3.5 
53.7 
44.2 

10.5 
10.2 
9.8 
9.5 

0  16.8 
0  20.0 
0  23.9 
0  28.2 
0  32.7 

3.2 
3.9 
4.3 
4.5 

4  52.6 
5     6.4 
5  20.3 
5  34.3 
5  48.4 

13.8 
13.9 
14.0 
14.1 

11  35.8 
11  46.3 
11  56.5 
12    6.3 
12  15.8 

10.5 
10.2 
9.8 
9.5 

13  43.2 
13  40.0 
13  36.1 
13  31.8 
13  27.3 

3.2 

3.9 
4.3 
4.5 

9     7.4 
8  53.6 
8  39.7 
8   25.7 
8  11.6 

13.8 
13.9 
14.0 
14.1 

9.0 

5.1 

14.2 

9.0 

5.1 

14.2 

20 
27 
28 
29 
30 

35.2 

26.5 
18.3 
10.4 
3.0 

8.7 
8.2 
7.9 
7.4 

0  37.8 
0  43.4 
0  49.5 
0  56.0 
1     3.0 

56 
6.1 
6.5 
7.0 

6     2.6   ,.  „  12  24.8 

6  16.9  :;«:13  33.5 

6  31.2  if*'?  1241.7 

6  45.6  Krji*  49-6 

7     0.0           |12  57.0 

8.7 
8.2 
7.9 
7.4 

13  22.2 
13   16.6 
13   10.5 
13     4.0 

12  57.0 

5.6 
6.1 
6.5 
7.0 

7  57.4 
7  43.1 
7  28.8 
7  144 
7     00 

14.3 
14.3 
144 

14.4 

TABLE  LIV.     Lunar  Nutation  in  Longitude. 
Argument.     Supplement  of  the  Node. 


0 

+ 

? 

Ills 

+' 

V* 

o 
0 

0.0 

8.5 

14.8 

17.3 

15.2 

8.8 

0 

30 

2 

0.6 

9.0 

15.1 

17.2 

14.9 

8.1 

28 

4 

1.2 

9.4 

15.4 

17.2 

14.5 

7.7 

26 

6 

1.7 

10.0 

15.6 

17.2 

14.2 

7.2 

24 

8 

2.3 

10.4 

15.9 

17.2 

13.8 

6.5 

22 

10 

2.9 

10.9 

16.4 

17.1 

13.5 

6.1 

20 

12 

3.5 

11.4 

16.3 

17.0 

13.0 

5.4 

18 

14 

4.1 

11.8 

16.5 

16.9 

12.6 

4.8 

16 

16 

4.6 

12.2 

16.7 

16.7 

12.2 

4.3 

14 

18 

5.2 

12.6 

16.8 

16.5 

11.8 

3.7 

12 

20 

5.8 

13.1 

16.9 

16.4 

11.3 

3.0 

10 

22 

6.2 

13.4 

17.1 

16.2 

10.9 

2.4 

8 

24 

6.9 

13.8 

17.1 

15.9 

10.4 

1.8 

6 

26 

7.4 

14.1 

17.2 

15.7 

9.8 

1.3 

4 

28 

7.8 

14.5 

17.2 

15.4 

9.4 

0.6 

2 

30 

8.5 

14.8 

17.3 

15.2 

8.8 

0.0 

0 

Xle 

xl 

IX* 

VIII* 

VII* 

Vis 

TABLE  LV.  81 

Moon's  Distance  from  the  North  Pole  of  the  Ecliptic 
Argument.     Supplement  of  Node  4- Moon's  Orbit  Longitude. 


III* 

IV* 

V* 

VI* 

VII* 

vm« 

84° 

85° 

OUT. 

for  10 

87° 

Diff. 
for  10 

89° 

Diff. 

for  10 

92° 

DiflT. 
for  10 

94° 

r     ' 

,    „ 

'    " 

„ 

,   ,. 

/    // 

/    // 

/    /, 

O      ' 

0  0 
30 
!  0 
30 
2  0 
30 

3916.0  2042.7 
39  16.7:22    4.2 
3918.8  2327.0 
3922.4  2451.0 
3927.3  2616.2 
3933.7.2742.6 

27.2 
27.6 
28.0 

28.4 
28.8 

1346.6 
16    6.9 
1827.8 
2049.5 
2311.8 
25  34.8 

46.8 
47.0 
47.2 
47.4 
47.7 

48   0.0 
5041.4 
5322.9 
56   4.3 
5845.7 

53.8 
53.8 
53.8 
53.8 
53.8 

2213.4  AC.  R 
«33.l£« 
2652.2?~  n 
29  10.2  4™ 
31  27.5^6 
33  44.2  4° 

1517.3 
1637.7 
1756.8 
1914.6 
2031.3 
21  46.7 

30  0 
30 
29  0 
30 
28  0 
30 

1  27  0 

29.2 

147.9 

53.8 

145.3 

3  0 

30 
4  0 
30 
5  0 

3941.5 
39  50.6 
40    1.2 
4013.2 
40  26.7 

2910.1 
3038.9 
32    8.8 
3339.9 
35  12.2 

29.fi 
30.0 
30.4 
30.8 

27  58.5 
30  22.8 
3247.7 
35  13.2 
3739.3 

48, 

48.3 
48.5 
48.7 

4   8.3 
649.5 
930.6 
1211.6 
1452.5 

53.7 
53.7 
53.7 
53.6 

36   02 

38  15.3 
4029.7 
4243.3 
44  56.2 

45.0 
44.8 
44.5 
44.3 

23  0.8 
24  13.7 
2525.3 
26  35.7 

2744.8 

27  0 
30 
26  0 
30 
25  0 

31.1 

48.9 

53.6 

44.0 

30 
6  0 
30 
7  0 
30 

4041.5 
4057.7 
41  15  4 
41  34.4 
41  54.8 

3645.6 
3820.1 
39  55.8 
41  32.7 
43  10.6 

31.5 
31.9 
32.3 
32.6 

40    6.1 
42  33.4 
45    1.2 
4729.6 
49  58.6 

49.1 
49.3 
49.5 
49.7 

17333 
20  14.0 
22  54.4 
25  34.8 
28  14.9 

53.6 
53.5 
53.5 
53.4 

47    8.1 
49  19.4 
51  29.7 
5339.3 
5548.0 

43.8 
43.4 
43.2 
42.9 

28  52.6 
2959.0 
31  4.3 
32  8.2 
33  10.9 

30 
24  0 
30 
23  0 
30 

33.0 

49.8 

53.3 

426 

8  0 
30 
9  0 
30 
10  0 

4216.7 
4239.9 
43    4.6 
43  30.6 
4358.1 

4449.7 
4629.9 
4811.2 
4953.5 
51  37.0 

33.4 
33.8 
34.1 
34.5 

5228.1 
5458.2 
5728.7 
5959.8 
231  3 

50.0 
50.2 
50.4 
50.5 

3054.9 
3334.7 
36  14.3 
3853.7 
41  32.8 

53.3 
53.2 
53.1 
53.0 

5755.8 

42.3 
42.0 
41.7 
41.5 

34  12.2 
35  12.2 
36  10.9 
37  8.3 
38  4.4 

22  0 
30 
21  0 
30 
20  0 

0   2.8 
2   8.9 
414.1 
618.4 

1 

34.9 

50.7 

53.0 

41.1 

30 
11  0 
30 
12  0 
30 

4426.9 

4457.1 
4528.8 
46    1.8 
4636.1 

5321.6 
55   7.1 
56  53.8 
5841.6 

35.2 
35.7 
35.9 
36.2 

5   3.3 

735.8 
10   8.8 
1242.1 
1516.0 

-Ofi  4411.7 

5?046504 

51  49287 
"•*  52    6.8 

51  '3,  54  44.6 

52.9 
52.8 
52.7 
52.6 

821.8 
1024.3 
1225.9 
1426.6 
1626.3 

40.8 
40.5 
40.2 
39.9 

3859.1 
39  52.5 
4044.6 
41  35.3 
4224.7 

30 
19  0 
30 

18  0 
30 

030  3 

36.6 

51.4' 

52.5 

J39.6 

13  0 
30 
14  0 
30 
15  0 

4711.9 
4749.0 
4827.5 
49    7.4 
49  48.7 

220.1 
4H.O 
6   2.9 
755.7 
949.6 

37.0 
37.3 
37.6 
38.0 

1750.2 
2024.9 
2259.9 
2535.3 
2811.1 

51.6 
51.7 
51.8 
51.9 

5722.1 
59  59.3 
236.2 
512.7 
748.9 

52.4 
52.3 
52.2 
52.1 

1825.0 
2022.8 
22  19.7 
2415.5 
26  10.4 

39.3 
38.0 
38.6 
38.3 

4312.7 
43  59.4 
4444.7 
4528.7 
4611.3 

17  0 
30 
16  0 
30 
15  0 

|84°       l86° 

88° 

91° 

93° 

94° 

II« 

I« 

0 

XI* 

X« 

IX* 

82  TABLE  LV. 

Moorfs  Distance  from  the  North  Pole  of  the  Ecliptic* 
Argument.     Supplement  of  Node+Moon's  Orbit  Longitude. 


Ill* 

IV. 

7* 

Via 

VII. 

vm« 

84° 

86° 

for  10 

88° 

DM. 

for  10 

91° 

Ditf. 

for  10 

93° 

Diti. 
for  10 

94° 

O    '         /      // 

/    ^ 

>    ,, 

,    ,, 

/       // 

/    ,/ 

O      ' 

15  0  49  48.7 
305031  3 
16  051  15.3 
3052    0.6 

949.6 
11  44.5 
13  40.3 

38.3 

38.6 
39.0 

28  11.1 
30  47.3 
33238 
36    0.7 

ft 

52.1 
52.2 
52.3 

Ro  A 

748.9 
10  24.7 
13    0.1 
15  35.1 

tt 

51.9 
51.8 
51.7 
R1  f\ 

26  10.4 
28    4.3 
29  57.1 
31  49.0 

38.0 
37.6 
37.3 
37  0 

46  11.3 
46  52.6 
47  32.5 
48  11.0 

15  0 
30 
14  0 
30 

17  05247.3 
30  53  35.3 

17  3o.O 
19  33.7 

39.6 

38  37.9 
41  15.4 

52.5 

18    9.8 
20  44.0 

51.4 

33  39.9 
35  29.7 

36.6 

4848.1 
49  23.9 

13  0 
30 

39.9 

52.6 

51.3 

36.2 

18  05424.7 
30  55  15.4 
19  056    7.5 
3057    0.9 
20  0  57  55.6 

21  33.4 
23  34.1 
25  35.7 
27  38.2 
29  41.6 

40.2 
40.5 
40.8 
41.1 

43  53.2 
4631.3 
49    9.6 
51  48.3 
54  27.2 

52.7 
52.8 
52.9 
53.0 

23  17.9 
2551.2 

28  24.2 
30  56.7 
33  28.7 

51.1 
51.0 
50.8 
50.7 

37  18.4 
39    6.2 
40  52.9 
42  38.4 
44  23.0 

35.9 
35.6 
35.2 
34.9 

49  58.2 
5031.2 
51  2.9 
51  33.1 
52  1.9 

12  0 
30 
11  0 
30 
10  0 

41.4 

53.0 

505 

34.5 

305851.7 
21  05949.1 
30    047.8 
22  0    1  47.8 
30    249.1 

31  45.9 
33  51.1 
35  57.2 
38    4.2 
40  12.0 

41.7 
42.0 
42.3 
42.6 

57    6.3 
59  45.7 

53.1 
53.2 
53.3 
53.3 

36    0.2 
3831.3 

41    1.8 
4331.9 
46    1.4 

50.4 
50.2 
50.0 
49.8 

46    6.5 
47  48.8 
49  30.1 
51  10.3 
52  49.4 

34.1 
33.8 
33.4 
33.0 

52  29.4 
52  55.4 
5320.1 
53  43.3 
54  5.2 

30 
9  0 
30 
8  0 
30 

225.3 
5    5.1 

745  1 

42.9 

53.4 

49.7 

32.6 

23  0    3  51.8 
30    455.7 
24  0    6    1.0 
30    7    7.4 
25  0    8  15.2 

42  20.7 
44  30.3 
46  40.6 
4851.9 
51    3.8 

43.2 
43.4 
43.6 
44.0 

10  25.2 
13    5.6 
1546.0 
18  26.7 
21    7.5 

53.5 
5*3.5 
53.6 
53.6 

48  30.4 
50  58.8 
53  26.6 
55  53.9 
58  20.7 

49.5 
49.3 
49.1 
48.9 

54  27.3 
56    4.2 
57  39.9 
59  14.4 

0  47  8 

32.3 
31.9 
31.5 
31.1 

5425.6 
54  44.6 
55  2.3 
55  18.5 
55  33.3 

7  0 
30 
6  0 
30 
5  0 

443 

536 

487 

30.8 

30    924.3 
26  0  10  34.7 
30  11  46.3 
27  0  12  59.2 
30  14  13.3 

53  16.7 
55  30.3 
57  44.7 
59  59.8 

44.5 
44.8 
45.0 
45.3 

23  48.4 
26  29.4 
29  10.5 
31  51.7 
34  33.0 

53.7 
53.7 
53.7 
53.7 

046.8 
3  123 
537.2 
8    1.5 
10  25.2 

48.5 
48.3 
48.2 
47.9 

220.1 
351.2 
521.1 
649.9 
8  17.4 

30.4 
30.0 
29.6 
29.2 

55  46.8 
55  58.8 
56  9.4 
56  18.5 
56  26.3 

30 
4  0 
30 
3  0 
30 

2  15.8 

45.6 

53.8 

47.7 

28.8 

28  0  1528.7 
30  16  45.4 
29  0  18    3.2 
301922.3 
30  0  20  42.7 

432.5 
649.8 
9    7.8 
11  26.9 
13  46.6 

45.8 
46.0 
46.4 
46.6 

37  14.3 
39  55.7 
42  37.1 
45  18.6 
48    0.0 

53.8 
53.8 
53.8 
53.8 

1248.2 
15  10.5 
17  32.2 
19  53.1 
22  13.4 

47.4 
47.2 
47.0 
46.7 

943.8 
11    9.0 
12  33.0 
13  55.S 
15  17.3 

28.4 
280 
27.6 

27.2 

56  32.7 
56  37.6 
56  41.2 
56  43.3 
56  44.0 

2  0 
30 
J  0 
30 
0  0 

85° 

87° 

89° 

92° 

94° 

94° 

If* 

I« 

0 

XI« 

X« 

IX* 

TABLE  LVI. 


83 


Equation  II  of  the  Moon's  Polar  Distance. 
Argument  II,  corrected. 


III*  j  diff. 

IV* 

diff. 

V* 

diff 

VI* 

diff 

VII* 

diff. 

VIII* 

diff. 

I 

0 

/. 

,  /• 

.    ,/ 

/    „ 

/    // 

f     rr 

0 

0 

1 

2 
3 
4 
5 

9  13.8 
9  13.9 
0  14.1 
0  14.5 
0  15.1 
0  15.8 

0.1 
0.2 

0.4 
0.6 
0.7 

1  24.4 
1  29.0 
1  33.8 
1  38.7 
1438 
149.0 

4.6 
4.8 
4.9 
5.1 
5.2 

436.9 
444.9 
4530 
5     1.1 
5    9.3 
5  17.6 

8.0 
8.1 
8.1 
8.2 
8.3 

9    0.0 
9    9.2 
9  18.4 

9  27.5  1 
9  36.7! 
9  45.9 

9.2 
9.2 
9.1 
9.2 
9.2 

1323.1 
1331.0! 
1338.8! 
1346.61 
1354.2! 
14    1.8 

7.9 

7.8 

7.6 
7.6 

16  35.6 
16  40.2 
1644.6 
1648.9 
16  53.0 
16  56.9 

4.6 
4.4 
4.3 
4.1 
3.9 

30 
29 
28 
27 
26 
25 

0.9 

5.3 

8.4 

9.1 

7.5 

3.8 

6 

7 
8 
9 
10 

0  16.7 
0  17.7 
0  18.9 
020.3 
021.8 

1.0 
1.2 
1.4 
1.5 

1  54.3 
1  59.8 
2  5.4 
2  11.1 
2  16.9 

5.5 
5.6 
5.7 

5.8 

526.0 
5  34  4 
542.9 
5  51.4 
6    0.0 

8.4 
8.5 
8.5 
8.6 

955.0 
10    4.1 
10  13.2 
10  22.3 
1031.4 

9.1 
9.1 
9.1 
9.1 

14    9.3  7  , 
14  16.7!  '•* 
1424.0   'I 
14  31.8  1'j; 

U38.2 

17  0.7 
17  4.4 
17  7.9 
17  11.3 
17  14.5 

3.7 
3.5 
3.4 
3.2 

24 
23 
22 
21 
20 

1.7 

6.0 

8.7 

9.0 

7.0 

3.0 

11 
12 
13 
14 
15 

023.5 
025.3 
027.3 
0  29.4 
031.7 

1.8 
2.0 
2.1 
2.3 

222.9 
229.0 
235.2 

J241.5 
12  47.9 

6.1 
6.2 
6.3 
6.4 

6    8.7 
6  17.4 
626.2 
635.0 
643.8 

8.7 

8.8 
8.8 
8.8 

10  40.4 
10  49.4 
10  58.4 
11    7.3 
11  16.2 

9.0!444'26.9 

90  14589  6S 
8.9  .,     ,.",.  6.6 
Q  q   15     5.5   fi  6 

89  15  12.1  *'* 

17  17.5 
17  20.4 
17  23.2 
17  25.8 
17  28.3 

2.9 
2.8 
2.6 
2.5 

19 

18 
17 
16 
15 

2.5 

6.6 

8.9 

8.8  | 

6.4 

2.3 

16 
17 
18 
19 
20 

0  34.2  „ 
036.81  £J| 
039.6  *  I 
048.5  rj 
045.5T 

254.5 
3  1.1 

3  7.9 
3  14.8 
321.8 

6.6 
6.8 
6.9 
7.0 

652.7 
7    1.6 
7  10.6 
7  10.6 
728.6 

8.9 
9.0 
9.0 
9.0 

11  25.0 
11  33.8 
11  42.6 
11  51.3 
12    0.0 

0S!1518.5 

ft  ft  15  24-8 
«7  1531.0 
a'  1537.1 
8-7|1543.1 

6.3 
6.2 
6.1 
6.0 

17  30.6 
17  32.7 
17347 
17  38.5 
17  38.2 

2.1 
2.0 
1  8 
1.7 

14 
13 
12 
11 
10 

3.2 

7.0 

9.1 

8.6  I 

5.8 

1.5 

21  048.7 
22  JO  52.1 

„  ,  3288 
34  3  36.0 

7.2 
7  ^ 

7  37.7  q 

746.8)^ 

12    8.6 
12  17.1 

8.5 

8C 

1548.9  *7 
1554.6!°-' 

17  39.7 
1741.1 

1.4 

i  t 

9 

8 

23 
24 
25 

,0  55.6 
059.3 
1    3.1 

3  7  3  43.3 
«Z  3  50.7 

O.O    Q    co  o 

o  56.2 

/  .0 

7.4 
7.5 

7  55.9 
8    5.0 
8  14.1 

«7.  A 

9.1 
9.1 

12  25.6 
12  34.0 
12  42.4 

.0 

8.4 
8.4 

16    0.2!?? 
16    5.7J  ll 
16  ll.ol5 

1742.3 
;  17  43.3 
1744.2 

1.0 
0.9 

7 
6 
5 

3.9 

7.6 

9.2 

8.3 

5.2 

0.7 

26 

27 
28 
29 
30 

1    7.0 
1  11.1 
1  15.4 
1  19.8 

1  24.4 

A  1    4     5"8 

ii  4  13  * 

4^21.2 
7'J  4  29.0 
4'6  4  36.9 

7.6 
7.8 
7.8 
7.9 

;8  23.3 
832.5 
841.6 
8  50.8 
9    0.0 

q  o  12  50.7 

"H*S 

q  „   Id     t.O 

'*•'*     1  Q    1  R   1 

Q  9  13  15.1 
'^13  23.1 

8.2 
8.1 
8.1 
8.0 

16  16.2  .  i  !l744.9 
16  21.3;  ?q   1745.5 
1626.2  ?  £  1745.9 
1631.0  7  £  1746.1 
1635.6  *     |1746.2 

0.6 
0.4 
0.2 
0.1 

4 
3 
2 
1 
0 

II* 

I* 

0 

XI* 

X*               IX* 

TABLE  LVII. 

Equation  III  of  Moorfs  Polar  Distance. 
Argument.     Moon's  True  Longitude. 


III* 

IV« 

V* 

VI* 

VII* 

VIII* 

i 

o 
0 

16.0 

14.9 

12.0 

8.0 

4.0 

1.1 

o 
30 

6 

16.0 

14.5 

11.3 

7.2 

33 

0.7 

24 

19 

15.8 

13.9 

10.5 

6.3 

2.6 

0.4 

18 

IS 

15.6 

13.4 

9.7 

5.5 

2.1 

0.2 

12 

24 

15.3 

12.7 

8.8 

4.7 

i.& 

0.0 

6 

30 

14.9 

12.0 

8.0 

4.0 

1.1 

0.0 

0 

II* 

1     I- 

0* 

XI* 

X* 

IX* 

84  TABLE  LVII1. 

To  convert  Degrees 
and  Minutes  into 
Decimal  Parts. 


TABLE  LIX. 

Equations  of  Moon's  Polar  Distance. 
Arguments,  Arg.  20  of  Long. ;  V  to  IX 

corrected ;  X  not  corrected ;  and  XI 

and  XII  corrected. 


Deg.  ,  Dec 
&Min.  parts. 

Arg 

20 

V 

VI 

VII 

vin 

IX 

X 

XI 

Arg 

Arg 

XII 

Arg. 

O   ' 

1  5 

003 

250 

0.3 

55.9 

6.1 

2.6 

25.1 

3.0 

0.7 

0.9 

850 

0 

4.0 

500 

1  26 

4 

2CO 

0.3 

55.8 

6.2 

2.725.1 

3.1 

0.7 

0.9 

240 

10 

3.7 

510 

148 

5 

270 

0.4 

55.7 

6.3 

2.8  25.0 

3.2  0.8 

1.0 

230 

20 

3.4 

520 

2  10 

6 

280 

0.6 

55.4 

6.5 

3.024.9 

3.5  1.0 

1.0 

220 

30 

3.1 

530 

231 

7 

290 

0.8 

55.1 

6.9 

3.324.8 

3.8  1.2 

1.1 

210 

40 

2.8 

540 

253 

8 

300 

1.0 

54.6 

7.3 

3.7  24.7 

4.3  1.5 

1.2 

200 

50 

2.5 

550 

3  14 

9 

310 

1.3 

54.1 

7.8 

4.224.4 

4.9  1.8 

1.3 

190 

60 

2.3 

500 

336 

10 

320 

1.7 

53.4 

8.4 

4.724.1 

5.6 

2.2 

1.4 

180 

70 

2.1 

570 

358 

11 

330 

2.1 

52.7 

9.1 

5.4  23.8 

6.4 

2.7 

1.5 

170 

801.9 

580 

4  19 

12 

340 

2.6 

51.9 

9.8 

6.1 

23.5 

7.2 

3.2 

1.7 

160 

90 

1.7 

590 

441 

13 

350 

3.1  'Sl.O 

10.7 

6.9 

23.2 

8.2 

3.8 

1.9 

150 

100 

1.6 

600 

5  2 

14 

360 

3.7 

50.0 

11.6 

7.7!  22.8 

9.2 

4.4 

2.1 

140 

110 

1.5 

610 

524 

15 

370 

4.3 

48.9 

12.6 

8.7'  22.4 

10.3 

5.1 

'2.3 

130 

120 

1.5 

620 

546 

16 

380 

4.9 

17.7 

13.6 

9.721.9 

11.5 

5.8 

2.5 

120 

130 

1.5 

630 

6  7 

17 

390 

5.6 

46.5 

14.8 

10.7 

21.4 

12.8 

6.6 

2.8 

110 

140 

1.5 

640 

629 

18 

400 

6.4 

45.2 

16.0 

11.8 

20.9 

14.1 

7.4 

3.0 

100 

150 

1.6 

650 

650 

19 

410 

7.1 

13.9 

17.2 

13.020.4 

15.5 

8.3 

3.3 

90 

160 

1.7 

660 

712 

20 

420 

7.9 

42.5 

18.5 

14.2 

19.9 

17.0 

9.1 

3.5 

80 

170 

1.9 

670 

734 

21 

430 

8.8 

41.0 

19.8 

15.5 

19.3 

18.5 

10.1 

3.8 

70 

180  2.1 

680 

755 

22 

440 

9.6 

39.5 

21.2 

16.8 

18.7 

20.1 

11.0 

4.1 

60 

190  2.3 

690 

817 

23 

450 

10.5 

38.0 

22.6  18.1 

18.1 

21.7 

12.0 

4.4 

50 

200 

2.5 

700 

838 

24 

460 

11.3 

36.4 

24.1 

19.4 

17.5 

23.3 

12.9 

4.7 

40 

210J2.8 

710 

9  0 

25 

470 

12.  2 

34.9 

25.5 

20.8 

16.9 

24.9 

13.9 

5.0 

30 

220  3.1 

720 

922 

26 

480 

13.2 

33.2 

27.0 

22.2 

16.3 

26.6 

15.0 

5.4 

20 

230)3.4 

730 

943 

27 

490 

14.1 

31.6 

28.5 

23.6 

15.6 

28.3 

16.0 

5.7 

10 

240  3.7 

740 

10  5 

28 

500 

150 

30.0 

30.0 

250 

15.0 

30.0 

17.0 

6.0 

0 

250 

4.0 

750 

1026 

29 

510  159 

284 

31.5 

26.414.4J31.7 

18.0 

6.3 

990 

260 

4.3 

760 

1048 

30 

520 

16.8 

26.8 

33.0  27.8 

13.7 

33.4 

19.0 

6.6 

980 

270 

4.6 

770 

11  10 

31 

530 

17.8 

25.1 

34.5  29.2 

13.1 

35.1 

20.1 

7.0 

970 

280 

49 

780 

11  31  32 

540 

18.7 

23.6 

35.9 

30.6 

12.5  36.7 

21.1 

7.3 

960 

290 

5.2 

790 

1153 

33 

550 

19.5 

22.0 

3T.4J31.9ll  1.9 

38.3 

22.0 

7.6 

950 

300 

5.5 

800 

1214 

34 

560 

20.4 

20.5 

38.8  33.2 

11.3 

39.9 

23.0 

7.9 

940 

310 

5.7 

810 

1236 

3ft 

570 

21.2 

19.0 

40.2  34.5  10.7 

41.5 

23.9 

8.2 

930 

320 

5.9 

820 

1258 

36 

580 

22.1 

17.5 

41.5  35.8 

10.1 

43.0 

24.9 

8.5 

920 

330 

6.1 

830 

1319 

37 

590 

22.9 

16.1 

42.8  37.0 

9.6 

44.5 

25.7 

8.7 

910 

340 

6.3 

840 

1341 

38 

600 

23.6 

14.8 

44.0  38.2 

9.1 

45.9 

26.6 

9.0 

900 

350 

6.4 

850 

14  2 

39 

610 

24.4 

13.5 

45.2  39.3 

8.6 

47.2 

27.4 

9.2 

890 

360 

6.5 

860 

1424 

40 

620 

25.1 

12.3 

46.4  40.3 

8.1 

48.5 

28.2 

9.5 

880 

370 

6.5 

870 

1446 

41 

630 

25.7 

11.1 

47.441.3 

7.6 

49.7 

28.9 

9.7 

870 

380 

6.5 

880 

15  7  42 

640  26.3 

10.0 

48.4  42.3 

7.2 

50.8 

29.6 

9.9 

860 

3UO 

6.5 

890 

1529 

43 

650  26.9 

9.0 

49.3  43.1 

6.8 

51.8 

30.2 

10.1 

850 

400 

6.4 

900 

1550 

44 

660  '27.4 

8.1 

50.2  43.9 

6.5 

52.8 

30.8 

10.3 

840 

410 

6.3 

910 

1612 

45 

670  !  27.  9 

7.3 

50.9  44.6 

6.2 

53.6 

31.3 

10.5 

830 

420 

6.1 

920 

1634 

46 

680  28.3 

6.6 

51.645.3 

5.9  54.4 

31.8 

10.6 

820 

430 

5.9 

930 

1655 

47 

690 

28.7 

5.9 

52.245.8 

5.6 

55.1 

32.2 

10.7 

810 

440 

5.7 

940 

1717 

48 

700 

29.0 

5.4 

52.7  46.3 

5.3 

55.7 

32.5 

10.8 

800 

450 

55 

950 

'1738 

49 

710 

29.2 

4.9 

53.1  '46.7 

5.2  56.2  32.8 

10.9 

790 

460 

5.2 

960 

18  0 

50 

720 

29.4 

4.6 

53.5  47.0 

5.1  56.5  33.0 

11.0  780 

470 

4.9 

970 

1822 

51 

730  29.6 

4.3 

53.7  47.2 

5.0  56.8  33.2 

11.0  770 

480 

4.6  980 

18  43  52 

740  !  29.7 

4.2 

53.8  47.3 

4.9  56.9  33.3 

11.1 

760 

490 

4.3  990  ! 

19  5  53 

750  [29.7 

4.1 

53.9  47.4 

4.9|57.033.3  11.1 

750 

500 

4.0  1000 

Constant  10" 


TABLE  LX.  TABLE  LXI.         35 

Small  Equations  of  Moorfs  Parallax.       Moons  Equatorial  Parallax. 
Args.,  1,  2,  4,  5,  6, 8, 9, 12, 13,  of  Long.         Argument.  Arg.  of  Evection. 


w 

2 

4      5 

6 

8 

9 

12 

13 

A 

00.0 

1.6 

0.6 

1.6 

1.9 

0.0 

3.6 

1.4 

2.0 

100 

30.0 

1.6 

0.6  1.6 

1.9 

0.0 

3.5 

1.4 

2.0 

97 

6  0.0 

1.5 

0.6   1.5 

1.8 

0.0 

3.1 

1.4 

1.9 

94 

I 

90.1 

1.5 

0.6   1.5 

1.8 

0.1 

2.6 

1.3 

.8 

91 

120.1 

1.4 

0.5   1.4 

1.7 

0.2 

1.9 

1.2 

.7 

88 

150.1 

1.3 

0.5  1.3 

1.6 

0.2 

1.3 

1.1 

.6 

85 

1 

180.2 

1.1 

0.4 

1.1 

1.4 

0.3 

0.7 

1.0 

.4 

82 

21  0.3 

1.0 

0.4 

1.0 

1.3 

0.5 

0.2 

0.9 

.2 

79 

240.4 

0.9  J0.3 

0.9 

1.2 

0.6 

0.0 

0.7 

.0 

76 

2705 

0.7  lo.3 

0.7 

1.0 

0.7 

0.1 

0.6 

0.9 

73 

30  0.5 

0.6  0.2 

0.6 

0.9 

0.8 

0.4 

0.5 

0.7 

70 

330.6 

0.4  10.2 

0.4 

0.7 

0.9 

0.8 

0.4 

0.5 

67 

3607 

0.3 

0.1 

0.3 

0.6 

1.0 

1.5 

0.3 

0.4 

64 

390.7 

0.2 

0.1 

0.2 

0.5 

1.1 

2.1  0.2 

0.2 

61 

42  0.8 

0.1 

0.0 

0.1 

0.4 

l.l 

2.8  0.1 

0.1 

58 

45  08 

0.0 

0.0 

0.0 

0.3 

1.2 

3.2  0.0 

0.0 

55 

480.3 

0.0  0.0 

0.0 

0.3 

1.2 

3.5  0.0 

0.0 

52 

500.8 

0.0  [0.0 

0.0 

0.3 

1.2 

3.6  0.0 

0.0 

60 

Constant    7" 

The  first  two  figures  only  of  the  Arguments 

are  taken. 

0* 

I* 

II* 

III* 

IV* 

V* 

o 

/     /, 

/    ,/ 

,    // 

„ 

„ 

„ 

0 

0 

1  20.8 

1  15.6 

1    1.5 

42.6 

24.1 

10.8 

30 

1 

1  20.8 

15.2 

1    0.9 

41.9 

23.6 

10.5  29 

2 

1  20.8 

14.9 

1    0.3 

41.3 

23.0 

10.2J28 

3 

1  20.7 

14.5 

59.7 

40.6 

22.5 

9.927 

4 

20.7 

14.2 

59.2 

40.0 

21.9 

9.6 

26 

5 

20.6 

13.8 

58.6 

39.4 

21.4 

9.4 

25 

6 

20.6 

1  13.4 

57.9 

38.7 

20.9 

9.1 

24 

7 

20.5 

1  13.0 

57.3 

38.1 

20.4 

8.8 

23i 

8 

20.4 

1  12.6 

56.7 

37.4 

19.9 

8.6 

22, 

9 

20.3 

1  12.2 

56.1 

36.8 

19.4 

8.4 

2? 

10 

20.2 

1  11.7 

55.5 

36.1 

18.9 

8.2 

20; 

11 

1  20.1 

1  11.3 

54.9 

35.5 

18.4 

8.0 

19' 

12 

1  19.9 

1  10.8 

54.2 

34.9 

17.9 

7.8 

id 

13 

14 

1  19.8 
1  19.6 

1  10.4 
1    9.9 

53.6 
53.0 

34.2 
33.6 

17.5 
17.0 

7.6 

7.4 

13 

15 

1  19.5 

1    9.4 

52.3 

33.0 

16.6 

7.2 

15 

16 

1  19.3 

1    9.0 

51.7 

32.4 

16.1 

7.1 

14 

17 

1  19.1 

1    8.5 

51.1 

Cl.7 

15.7 

6.9 

13 

18 

1  18.9 

1    8.0 

50.4 

31.1 

15.2 

6.S 

12 

19 

18.7 

1    7.5 

49.8 

30.r» 

14.8 

6.7 

11 

20 

18.4 

1    7.0 

49.1 

29.S 

14.4 

6.5 

10 

21 

18.2 

1    6.5 

48.5 

29.3  14.0 

6.4 

9 

22 

18.0 

1    5.9 

47.8 

28.7  13.6 

6.3 

8 

23 

17.7 

1    5.4 

47.2 

28.1 

13.2 

6.3 

1 

24 

17.4 

1    4.8 

46.5 

27.5 

«2.9 

6.2 

6 

25 

17.1 

1    4.3 

45.9 

26.9 

12.5 

6.1 

5 

26 

16.9 

1    3.8 

45.2 

26.3 

12.1 

6.1 

4 

27 

16.6 

1    3.2 

44.6  25.8 

11.8 

6.1 

3 

28 

16.2 

1    2.6 

43.9  25.2 

11.5|  6.0 

2 

29 

15.9 

1    2.1 

43.3124.7 

11.1 

6.0 

I 

30     15.6J1    1.5 

42.6;24.l|l0.8 

6.0    0 

XI* 

X* 

IX* 

vinJvii* 

VI* 

34 


86 


TABLE  LXII. 
Moon's  Equatorial  Parallax. 

Argument.      Anomaly. 


0     diff|     I* 

diff     II* 

diff 

III* 

diff 

IV* 

diff      V«     diff 

0 

'    "    \- 

/    /•/ 

,    „ 

/    „ 

,    „ 

/    ,/ 

o 

0 

1 

2 
3 
4 
5 

58  57.7! 
5857.7 
58  57.6 
58  57.4 
5857.1 
5856.8 

0.0 
0.1 
0.2 
0.3 
0.3 

58  27.0 
5825.0 
5823.0 
5820.9 
5818.7 
5816.5 

«  S  I! 

,-W.U        cry           i       p 

22  5658'4 
~;  5655.2 

•  !5652.0 

3.1 
3.2 
3.2 
3.2 
3.2 

5529.8 
5526.6 
5523.4 
5520.2 
55  17.0 
55  13.8 

3.2 
3.2 
3.2 
3.2 
3.2 

54   1.9 
53  59.4 
5356.9 
5354.5 
5352.1 
5349.7 

2.5 

2.5 
2.4 
2.4 
2.4 

53    3.2  " 
53    1.8  J;J 
53    0.5  \* 
5259.3}'* 

5258.1-2 
52  57.0.  1'1 

30 

29 
28 
27 
26 
25 

0.4 

2.2 

3.2 

3.2 

2.3 

1.2 

6 

7 
8 
9 
10 

58  56.4 
5856.0 
58  55.4 
58  54.8 
58  54.2 

0.4 
0.6 
0.6 
0.6 

58  14.3 
58  12.  0 
58    9.6 
58    7.2 
58    4.8 

23 

2.4 
2.4 
2.4 

5648.8 
5645.5 
56  42.3 
56  39.0 
5635.7 

3.3 
3.2 
3.3 
3.3 

5510.6 
55    7.5 
55   4.4 
55    1.3 
5458.2 

3.1 
3.1 
3.1 
3.1 

5347.4 
5345.1 
5342.9 
53406 
53  38.5 

2.3 
2.2 
2.3 
2.1 

52  55.8  in 
5254.8  'J-JJ 

58  53.8  ['J 

5251.9;0'9 

24 
23 

22 
21 
20 

0.8 

2.5 

3.3 

3.1 

2.2 

10.9 

11 
12 

58  53.4 
5852.6 

0.8 

58    2  3 
5759.8^1 

5632.4 
5629.1 

3.3 

5455.1 

5452.1 

3.0 

5336.3 
53  34.2 

2.1 

5251.01 
5250.1  "'£ 

19 
18 

13 

5851.8 

0.8 

1  o 

5757.2*'° 

56  25.8 

3.3 

O  Q 

5449.1 

3.0 

O  A 

5332.1 

2.1 

O  A 

5249.3^ 

17 

14 
15 

58  50.8 
58  49.8 

l.U 

1.0 

5754.6^ 
5751.  9  <2'7 

56  22.5 
56  19.2 

o.o 

3.3 

5446.1 
5443.1 

o.O 

3.0 

5330.1 
5328.1 

Z.O 

2.0 

5248.6 
52  47.9 

16 
15 

1.1 

2.7 

3.3 

2.9 

1.9 

0.7 

16 
17 
18 
19 
20 

5848.7 
58  47.6 
58  46.4 
5845.1 
5843.8 

1.1 
1.2 
1.3 
1.3 

5749.2 
5746.4!^'° 
57  43.7  ~7 
5740.8J~-£ 
57  38.0  *8 

56  15.9 
5612.6 
56    9.3 
56    6.0 
56   2.7 

3.3 
3.3 
3.3 
3.3 

5440.2 
5437.3 
5434.4 
5431.5 
5428.7 

2.9 
2.9 
2.9 

2.8 

5326.2 
53  24.3 
53  22.4 
5320.6 
53  18.8 

1.9 
1.9 
1.8 
1.8 

52  47.2 
5246.6 
5246.0 
5245.5 
52  45.0 

0.6 
0.6 
0.5 
0.5 

14 
13 

12 
11 
10 

1.4 

2.9 

3.4 

2.8 

1.8 

0.4 

21 
22 
23 
24 
25 

58  42.4 
5840.9 
5839.4 
58  37.8 
5836.2 

1.5 
1.5 
1.6 
1.6 

57  35  1 
57  32.2 
5729.3 
5726.3 
5723.3 

2.9 
2.9 
3.0 
3.0 

5559.3 
55  56.0 
55  52.7 
55  49.4 
5546.1 

3.3 
3.3 
3.3 
3.3 

5425.9 
5423.1 
5420.3 
5417.6 
54  14.9 

2.8 
2.8 
2.7 
2.7 

5317.0 
53  15.3 
53  13.7 
53  12.0 
53  10.4 

1.7 

1.6 
.7 
.6 

52  44.6 
5244.2 
52  43.8 
52  43.5 
5243.3 

0.4 

0.4 
0.3 

0.2 

9 
8 
7 
6 
5 

1.8 

3.0 

3.3 

2.7 

.5 

0.2 

26 
27 
28 
29 
30 

58  34.4 
58  32.7 
58  30.9 
5829.0 
5827.0 

1.7 
1.8 
1.9 
20 

5720.2 
57  17.2 
5714.1 
5711.0 
57  7.9 

3.0 
3.1 
3.1 
31 

5542.8 
5539.6 
55  36.4 
5533.1 
5529.8 

3.2 
3.2 
3.3 
3.3 

54  12.2 
54   9.6 
54   7.0 
54  4.4 
54    1.9 

2.6 
2.6 
2.6 
2.5 

53   8.9 
53   7.4 
53   5.9 
53   4.5 
53   3.2 

.5 

A 
3 

5243.1 

5242.9 
5242.8 
5242.7 
5242.7 

0.2 
0.1 
0.1 
0.0 

4 
3 
2 

1 
0 

XI* 

X« 

IX* 

VIII* 

VII* 

VI* 

I 

TABLE   LXIII. 


87 


Moorfs  Equatorial  Parallax. 
Argument.     Argument  of  the  Variation. 


0« 

I* 

n« 

III* 

IV* 

v« 

0 

0 

55.6 

42.3 

// 

16.0 

3.7 

17.6 

// 

44.0 

o 
30 

1 

55.6 

41.5 

15.3 

3.8 

18.5 

44.8 

29 

2 

55.5 

40.7 

14.5 

3.8 

19.3 

45.6 

28 

3 

55.5 

39.8 

13.8 

3.9 

20.1 

46.3 

27 

4 

55.3 

39.0 

13.1 

4.1 

21.0 

47.0 

26 

5 

55.2 

38.1 

12.4 

4.3 

21.9 

47.7 

25 

6 

55.0 

37.2 

11.7 

4.5 

22.7 

48.4 

24 

7 

54.8 

36.3 

11.1 

4.7 

23.6 

49.1 

23 

8 

54.6 

35.5 

10.4 

5.0 

24.5 

49.7 

22 

9 

54.3 

34.6 

9.8 

5.3 

25.4 

50.3 

21 

10 

54.0 

33.7 

9.2 

5.6 

26.3 

50.9 

20 

11 

53.7 

32.7 

8.7 

6.0 

27.2 

51.5 

19 

12 

53.3 

31.8 

8.2 

6.3 

28.2 

52.1 

18 

13 

52.9 

30.9 

7.7 

6.8 

29.1 

52.6 

17 

14 

52.5 

30.0 

7.2 

7.2 

30.0 

53.1 

16 

15 

52.0 

29.1 

6.7 

7.7 

30.9 

53.5 

15 

16 

51.5 

28.2 

6.3 

8.2 

31.8 

54.0 

14 

17 

51.0 

27.2 

5.9 

8.7 

32.8 

54.4 

13 

18 

50.5 

26.3 

5.6 

9.3 

33.7 

54.8 

12 

19 

49.9 

25.4 

5.3 

9.8 

34.6 

55.1 

11 

20 

49.4 

24.5 

5.0 

10.5 

35.5 

55.4 

10 

21 

48.8 

23.6 

4.7 

11.1 

36.4 

55.7 

9 

22 

48.1 

22.7 

4.5 

11.7 

37.3 

56.0 

8 

23 

47.4 

21.9 

4.3 

12.4 

38.2 

56.2 

7 

24 

46.8 

21.0 

4.1 

13.1 

39.0 

56.4 

6 

25 

46.1 

20.1 

3.9 

13.8 

39.9 

56.6 

5 

26 

45.4 

19.3 

3.8 

14.5 

40.8 

56.8 

4 

27 

44.6 

18.5 

3.7 

15.3 

41.6 

56.9 

3 

28 

43.9 

17.6 

3.7 

16.1 

42.4 

56.9 

3 

29 

43.1 

16.8 

3.7 

16.8 

43.2 

57.0 

1 

30 

42.3 

16.0 

3.7 

17.6 

44.0 

67.0 

0 

XI« 

X» 

IX« 

VIII* 

VII« 

VI« 

88       TABLE  LXIV. 


TABLE  LXV. 


Reduction  of  the  Parallax, 
and  also  of  the  Latitude. 

Argument.     Latitude. 


Moon's  Semi-diameter. 
Argument.     Equatorial  Parallax. 


Lat 

Red. 

Red.  of 

T  at 

Eq.Par  Semidia. 

Eq.  Parj  Semidia 

Eq.Pai 

Semidia 

seel  Pro. 
Par. 

o 

Ol  pill 

JUcLl. 

53     0 

» 

26.5 

56 

0 

15  15.6 

59     0 

16 

4.6 

1 

0.3 

0 

0.0 

0    0.0 

53  10 

I  14 

29.3 

58  10 

15   18.3 

59   10 

16 

7.4 

2 

0.5 

3 

0.0 

1  11.8 

53  20 

14 

32.0 

56  20 

15  21.0 

59  20 

16  10.1 

3 

0.8 

6 

0.1 

2  22.7 

53  30 

14 

34.7 

56  30 

15  23.8 

59  30 

16  12.8 

4 

1.1 

9 

0.3 

3  32.1 

53  40    14 

37.4 

56  40 

15  26.5 

59  40 

16  15.6 

5 

1.4 

12 
15 

0.5 
0.7 

4  39.3 
5  43.4 

53  50 
54     0 

14 
14 

40.2 
42.9 

56  50 
57     0 

15  29.2 
15  31.9 

59  50 
60     0 

16   18.3 
16  21.0 

6 

7 

1.6 
1.9 

18 

1.0 

6  43.7 

54  10 

14 

45.6 

57  10 

15  34.7 

60   10 

16  23.7 

8 

2.2 

21 

1.4 

7  39.7 

54  20 

14 

48.3 

57  20 

15  37.4 

60  20 

16  26.4 

9 

2.4 

24 

1.8 

8  30.7 

54  30 

14 

51.1 

57  30 

15  40.1 

no  30 

16  29.2 

10 

2.7 

27 
3D 

2.3 
2.7 

9  16.1 
9  55.4 

54  40 
54  50 

14 
14 

53.8 
56.5 

57  40 
57  50 

15  42.8 
15  45.6 

60  40 
60  50 

16  31.9 
16  34.6 

33 

3.3 

10  28.3 

55     0 

14 

59.2 

58 

0 

15  48.3 

61     0 

16  37.3 

36 

3.8 

10  54.3 

55  10 

15 

2.0 

58  10 

15  51.0 

61    10 

16  40.1 

39 

4.4 

11  13.2 

55  20 

15 

4.7 

58  20 

15  53.7 

61   20 

16  42.8 

42 
45 

4.9 
5.5 

11  24.7 
11  28.7 

55  30 
55  40 

15 
15 

7.4 
10.1 

58  30 
58  40 

15  56.5 
15  59.2 

61  30 
61  40 

16  45.5 
16  48.2 

48 

6.1 

11  25.2 

55  50 

15 

12.9 

58  50 

16     1.9 

61  50 

16  51.0 

51 

6.7 

11  14.1 

56     0 

15 

15.6 

59 

0  1  16     4.6 

62     0 

16  53.7 

54 
57 

7.2 
7.8 

10  55.7 
10  30.0 

60 

8.3 

9  57.4 

63 

8.8 

9  18.3 

66 

9.2 

8  32.9 

69 

9.7 

7  42.0 

TABLE  LXVI. 

72 

10.0 

6  45.9 

75 

10.3 

5  45.4 

Augmentation 

of  Moon's  Semi-diameter. 

78 

10.6 

4  41.0 

81 

10.8 

3  33  5 

84 

1LO 

2  23.7 

A  It 

Horizon.  Semi-diameter 

Horizon.    Semi-diameter. 

87 
90 

11.1 
11.1 

1  12.3 
0    0.0 

All, 

14'30" 

15' 

16' 

17 

14'  30" 

15' 

16       17 

Subsidiary  Table. 

O 

2 

0.6 

0.6 

0.7 

0.8 

42 

9.2 

9.8 

11.2    12.6 

Lat. 

+  3' 

—  3' 

4 

1 

.0 

1.1 

1.3 

1.5 

45 

9.7 

10.4 

11.8    13.3 

6 

1 

fi 

1.6 

1.9 

2.1 

48 

10.2 

10.9 

12.4    14.0 

o 

rf 

" 

8 

2.0 

2.1 

2.4 

2.7 

51 

10.6 

11.4 

13.0    14.7 

0 

+  0.0 

—  0.0 

10 

2.4 

2.6 

3.0 

3.4 

54 

11.1 

11.8 

13.5    15.2 

6 

0.0 

0.0 

12 

0.0 

0.0 

12 

2.9 

3.1 

3.6 

4.0 

57 

11.5 

123 

14.0    15.8 

15 

0.0 

0.0 

14 

3.4 

3.6 

4.1 

4.7 

60 

11  8 

12.7 

14.4    16.3 

18 

0.1 

0.1 

16 

3.8 

4.1 

4.7 

5.3 

63 

12.2 

13.0 

14.9    16.8 

24 

18 

4.3 

4.6 

5.2 

5.9 

66 

12.5 

13.4 

15.2    17.2 

' 

' 

21 

4.9 

5.3 

6.0 

6.8 

69 

12.8 

13.7 

15.6    17.6 

30 

0.1 

0.1 

36 

0.2 

0.2 

24 

5.6 

6.0 

6.8 

7.7 

72 

13.0 

13.9 

15.9    17.9 

42 

0.2 

0.2 

27 

6.2 

6.7 

7.6 

8.6 

75 

13.2 

14.1 

16.1     18.2 

48 

0.3 

0.3 

30 

6 

9 

7.3 

8.4 

9.5 

78 

13.4 

14.3 

16.3    18.4 

54 

0.3 

0.3 

33 

7 

5 

8.0 

9.1 

10.3 

81 

13.5 

14.4 

16.5    18.6 

36 

8 

1 

8.6 

9.8 

11.1 

84 

13.6 

14.5 

16.6    18.7 

60 

79 

0.4 

n  i 

0.4 

A   K. 

39 

8.6 

9.2 

10.5 

11.9 

90 

13.7 

14.6 

16.7    18.8 

Im 

78 

u.o 
0.6 

u.o 
0.6 

84 

0.6 

0.6 

90 

4-0.6    —0.6 

TABLE   LXVII. 


89 


Moon's  Horary  Motion  in  Longitude. 
Arguments.  1  to  18  of  Longitude. 


Alt 

2 

3 

4 

5 

6 

1 

7 

8 

9 

Arg. 

0 

100 

5.0 

0.0 

2.9 

1.9 

0.0 

0.00 

0.00 

0.00 

0.16 

2 

5.0 

0.0 

2.8 

1.9 

0.0  0.00 

0.00 

0.00 

0.15 

98 

4 

4.9 

0.0 

2.8 

1.9 

0.0  0.01  0.00 

0.02 

0.15 

96 

G 

4.8 

0.1 

2.8 

1.9 

0.1  0.03  0.01 

0.05 

0.14 

94 

8 

4.7 

02 

2.7 

1.8 

0.1 

0.0610.01 

0.09 

0.12 

92 

10 

45 

0.3 

2.6 

1.7 

0.2 

0.09  0.02 

0.14 

0.10 

90 

12 

4.3 

0.4 

2.5 

1.7 

0.2 

0.1310.02 

0.19 

0.09 

88 

14 

4.1 

0.6 

23 

1.6 

0.3 

0.18 

0.03 

0.26 

0.07 

86 

16 

3.8 

0.7 

2.2 

1.5 

0.4 

0.23 

0.04 

0.33 

0.05 

84 

18 

3.6 

0.9 

2.0 

1.4 

0.5 

0.28 

0.05 

0.41 

0.03 

82 

20 

3.3 

1.1 

1.9 

1.3 

0.6 

0.34 

0.06 

0.50 

0.02 

80 

22 

3.0 

1.3 

1.7 

1.1 

0.7 

0.40 

0.07 

0.58 

0.01 

78 

24 

2.7 

1.5 

1.5 

1.0 

0.8 

0.46 

0.08 

0.67 

0.00 

76 

26 

2.3 

1.7 

1.3 

0.9 

0.9 

0.52 

0.10 

0.77 

000 

74 

28 

2.0 

1.9 

1.2 

0.8 

1.0 

0.58 

0.11 

0.86 

0.00 

72 

30 

1.7 

2.1 

1.0 

0.7 

1.1 

0.63 

0.12 

0.94 

0.01 

70 

32 

1.4 

2.2 

0.8 

0.5 

1.2 

0.69 

0.13 

.03 

0.01 

68 

34 

1.2 

2.4 

0.7 

0.4 

1.3 

0.74 

0.14 

.11 

003 

66 

36 

0.9 

2.6 

0.5 

0.3 

1.3 

0.78 

0.15 

.18 

0.05 

64 

33 

0.7 

2.7 

0.4 

0.3 

1.4 

0.82 

0.16 

.25 

0.06 

62 

40 

0.5 

2.8 

0.3 

0.2 

1.5 

0.86 

0.16 

.30 

0.08 

60 

42 

0.3 

2.9 

0.2 

0.1 

1.5 

0.89 

0.17 

1.35 

0.10 

58 

44 

0.2 

3.0 

0.1 

0.1 

1.6 

0.91 

0.17 

1.39 

0.11 

56 

46 

0.1 

3.1 

0.0 

0.0 

1.6 

0.93 

0.18 

1.42 

0.12 

54 

48 

0.0 

3.1 

0.0 

0.0 

1.6 

0.94 

0.18 

1.44 

0.13 

52 

50  0.0 

3.1 

0.0 

0.0 

1.6 

0.94 

0.181  1.44 

0.13 

50 

Arg. 

10 

11   12 

13   14 

15 

16 

17 

18 

Alg. 

0 

100 

0.00 

0.26 

000 

0.00 

0.00 

0.00 

0.26 

0.00 

0.21 

2 

0.00 

0.25  0.00 

o  oo  o.oo  ;  o.oo 

0.26 

0.00 

0.20 

98 

4 

0.02 

0.24  j  0.01 

ooo;o.oi  io.oo 

O.C6 

0.00 

0.20 

96 

6 

0.04 

0.22  0.03 

0.01 

O.C2  0.01 

0.25 

0.00 

0.20 

94 

8 

0.08 

0.20 

0.04 

0.02 

0.0410.01 

025 

0.01 

0.20 

92 

10 

0.12 

0.17 

0.07 

0.03 

0.06 

0.02 

0.24 

0.01 

0.20 

90 

12 

0.16 

0.14 

0.09 

0.04 

0.09 

0.02 

0.22 

0.02 

0.19 

88 

14 

0.20 

0.11 

0.12 

0.06 

0.12 

0.03 

0.21 

0.02 

0.19 

86 

16 

0.24 

0.08 

0.16 

0.07 

0.15 

0.04 

0.20 

0.03 

0.18 

84 

18 

0.28 

0.05 

0.19 

0.09 

0.19 

0.05 

0.19 

0.04 

0.18 

82 

20 

0.31 

0.03 

0.23 

0.11 

0.22 

0.06 

0.17 

0.05 

0.17 

80 

22 

0.34 

0.01 

0.27 

0.13 

0.26 

0.07 

0.15 

0.06 

0.17 

78 

24 

0.35 

0.00 

0.31 

0.15 

0.30 

0.08 

0.14 

0.07 

0.16 

76 

26 

036 

0.00 

035 

0.17 

0.34 

0.08 

0.12 

0.07 

0.16 

74 

28 

035 

0.01 

0.39 

0.19 

0.38 

0.09 

0.11 

0.08 

0.15 

72 

30 

0.34 

0.02 

0.43 

0.21 

0.42 

0.10 

0.09 

0.09 

0.15 

70 

32 

0.32 

0.04 

0.47 

0.23 

0.45 

0.11 

0.07 

0.10 

0.14 

68 

34 

0.29 

0.06 

0.50 

0.25 

0.49 

0.12 

0.06 

0.11 

0.14 

66 

36 

0.26 

0.09 

0.54 

0.26 

0.52 

0.13 

0.05 

0.12 

0.13 

64 

38 

0.22 

0.11 

0.57 

0.28 

0.55 

0.14 

0.04 

0.12 

0.13 

62 

40 

0.18 

0.14 

0.59 

0.29 

0.58 

0.14 

0.02 

0.13 

0.12 

60 

42 

0.15 

0.16 

0.62 

0.30 

0.60 

0.15 

0.01 

0.13 

0.12 

58 

44 

0.12 

0.19 

0.63 

0.31  0.62 

0.15 

0.01 

0.14  0.12 

56 

46 

0.10 

021 

0.65 

0.32  0.63 

0.16- 

0.00 

0.14  0.12 

54 

48  0.09 

0.22 

0.66 

0.32  0.64 

0.16 

0.00 

O.H  "  12 

52 

50  0.08 

0.22  0.66 

0.32  0.64 

0.16  0.00  0.14(0.11 

50 

90  TABLE  LXVIII. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Argument  of  the  Evection. 


0* 

I« 

II* 

III* 

IV* 

V* 

0 

// 

„ 

// 

// 

// 

// 

o 

0 

80.3 

74.7 

59.6 

39.4 

198 

r>.9 

30 

1 

80.3 

74.3 

58.9 

38.7 

19.3 

5.6 

29 

2 

80.3 

73.9 

58.3 

38.0 

18.7 

5.3 

28 

3 

80.2 

73.5 

57.7 

37.3 

18.1 

5.0 

27 

4 

80.2 

73.1 

57.1 

36.6 

17.6 

4.7 

26 

5 

80.1 

72.7 

56.4 

36.0 

17.0 

4.4 

25 

6 

80.1 

72.3 

55.8 

35.3 

16.5 

4.1 

24 

7 

80.0 

71.9 

55.1 

34.6 

15.9 

3.8 

23 

8 

79.9 

71.4 

54.5 

33.9 

15.4 

3.6 

22 

9 

79.8 

71.0 

53.8 

33.2 

14.9 

3.4 

21 

10 

79.7 

70.5 

53.1 

32.5 

14.4 

3.1 

20  | 

11 

79.5 

70.1 

52.5 

31.9 

13.9 

2.9 

19 

12 

79.4 

69.6 

51.8 

31.2 

13.4 

2.7 

18 

13 

79.2 

69.1 

51.1 

30.5 

12.9 

2.5 

17 

14 

79.1 

68.6 

50.5 

29.9 

12.4 

2.3 

16 

1.5 

78.9 

68.1 

49.8 

29.2 

11.9 

2.1 

15 

16 

78.7 

67.6 

49.1 

28.6 

11.4 

2.0 

14 

17 

78.5 

67.0 

48.4 

27.9 

11.0 

.8 

13 

18 

78.2 

66.5 

47.7 

27.2 

10.5 

.7 

12 

19 

78.0 

66.0 

47.0 

26.6 

10.1 

.6 

11 

20 

77.8 

65.4 

46.4 

26.0 

9.7 

.4 

10 

21 

77.5 

64.9 

45.7 

25.3 

9.3 

.3 

9 

22 

77.2 

64.3 

45.0 

24.7 

8.8 

.2 

8 

23 

77.0 

63.7 

44.3 

24.1 

8.4 

.2 

7 

24 

70.7 

63.2 

43.6 

23.5 

8.0 

.1 

6 

25 

76.4 

62.6 

42.9 

22.8 

7.7 

.0 

5 

26 

76.1 

62.0 

42.2 

22.2 

7.3 

1.0 

4 

27 

75.7 

61.4 

41.5 

21.6 

6.9 

0.9 

3 

28 

75.4 

60.8 

40.8 

21.0 

6.6 

0.9 

2 

29 

75.0 

60.2 

40.1 

20.4 

6.2 

0.9 

1 

30 

74.7 

59.6 

39.4 

19.8 

5.9 

0.9 

0 

XI* 

X* 

IX* 

VIII* 

VII* 

VI* 

TABLE  LXIX. 

Moon's  Horary  Motion  in  Longitude. 

Arguments.     Sum  of  Equations,  2,  3,  &c.,  and  Evection  scrrected 
!  I    0"  I    10"  I    20"  I  i 


s         ° 

a     o 

0       0 

0.0 

0.2 

0.5 

XII     0 

I         0 

0.0 

0.2 

0.4 

XI      0 

II       0 

0.1 

0.2 

0.3 

X       0 

III      0 

0.2 

0.2 

0.2 

IX      0 

IV      0 

0.3 

0.2 

0.1 

VIII   0 

V       0 

0.4 

0.2 

0.0 

VII     0 

VI      0 

0.5 

0.2 

0.0 

VI      0 

0" 

10" 

20" 

TABLE   LXX. 


91 


Moorfs  Horary  Motion  in  Longitude. 
Arguments.     Sum  of  preceding  equations,  and  Anomaly  corrected 


°"  1 

10" 

20" 

30" 

40" 

50" 

60" 

70" 

80" 

90" 

100" 

8         ° 

~~7> 

~ 

~ 

/, 

~~7, 

~ 

~ 

// 

// 

~7> 

// 

8    o 

0     0 

4.1 

5.3 

6.5 

7.6 

8.8 

10.0 

11.2 

12.4 

13.5 

14.7 

15.9 

XII    0 

5 

4.1 

5.3 

6.5 

7.7 

8.8 

10.0 

11.2 

12.3 

13.5 

14.7 

15.9 

25 

10 

4.2 

5.4 

6.5 

7.7 

8.8 

10.0 

11.2 

12.3 

13.5 

14.6 

15.8 

20 

15 

4.3 

5.5 

6.6 

7.7 

8.9 

10.0 

11.1 

12.3 

13.4 

14.5 

15.7 

15 

20 

4.5 

5.6 

6.7 

7.8 

8.9 

10.0 

11.1 

12.2 

13.3 

14.4 

15.5 

10 

25 

4.8 

5.8 

6.9 

7.9 

9.0 

10.0 

11.0 

12.1 

13.1 

14.2 

15.2 

5 

I       0 

5.1 

6.0 

7.0 

8.0 

9.0 

10.0 

11.0 

12.0 

13.0 

14.0 

14.9 

XI     0 

5 

5.4 

6.3 

7.2 

8.2 

9.1 

10.0 

10.9 

11.8 

12.8 

13.7 

14.6 

25 

10 

5.7 

6.6 

7.4 

8.3 

9.2 

10.0 

10.8 

11.7 

12.6 

13.4 

14.3 

20 

15 

6.1 

6.9 

7.7 

8.5 

9.2 

10.0 

10.8 

11.5 

12.3 

13.1 

13.9 

15 

20 

6.6 

7.2 

7.9 

8.6 

9.3 

10.0 

10.7 

11.4 

12.1 

12.8 

13.4 

10 

25 

7.0 

7.6 

8.2 

8.8 

9.4 

10.0 

10.6 

11.2 

11.8 

12.4 

13.0 

5 

II     0 

7.5 

8.0 

8.5 

9.0 

9.5 

10.0 

10.5 

11.0 

11.5 

12.0 

12.5 

X       0 

5 

7.9 

8.4 

8.8 

9.2 

9.6 

10.0 

10.4 

10.8 

11.2 

11.6 

12.1 

25 

10 

8.4 

8.7 

9.1 

9.4 

9.7 

10.0 

10.3 

10.6 

10.9 

11.3 

11.6 

20 

15 

8.9 

9.1 

9.4 

9.6 

9.8 

10.0 

10.2 

10.4 

10.6 

10.9 

11.1 

15 

20 

9.4 

9.5 

9.7 

9.8 

9.9 

10.0 

10.1 

10.2 

10.3 

10.5 

10.6 

10 

25 

9.9 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.1 

10.1 

5 

III    0 

10.4 

10.3 

10.2 

10.1 

10.1 

10.0 

9.9 

9.9 

9.8 

9.7 

9.6 

IX      0 

5 

10.8 

10.7 

10.5 

10.3 

10.2 

10.0 

9.8 

9.7 

9.5 

9.3 

9.2 

55 

10 

11.3 

11.0 

10.8 

10.5 

10.3 

10.0 

9.7 

9.5 

9.2 

9.0 

8.7 

20 

15 

11.7 

11.4 

11.0 

10.7 

10.3 

10.0 

9.7 

9.3 

9.0 

8.6 

8.3 

15 

20 

12.1 

11.7 

11.3 

10.9 

10.4 

10.0 

9.6 

9.1 

8.7 

8.3 

7.9 

10 

25 

12.5 

12.0 

11.5 

11.0 

10.5 

10.0 

9.5 

9.0 

8.5 

8.0 

7.5 

5 

IV    0 

12.9 

12.3 

11.7 

11.2 

10.6 

10.0 

9.4 

8.8 

8.3 

7.7 

7.1 

VIII  0 

5 

13.3 

12.6 

11.9 

11.3 

10.6 

10.0 

9.4 

8.7 

8.1 

74 

6.7 

25 

10 

13.6 

12.9 

12.1 

11.4 

10.7 

10.0 

9.3 

8.6 

7.9 

7.1 

6.4 

20 

15 

13.9 

13.1 

12.3 

11.5 

10.8 

10.0 

9.2 

8.5 

7.7 

6.9 

6.1 

15 

20 

14.1 

13.3 

12.5 

11.6 

10.8 

10.0 

9.2 

8.4 

7.5 

6.7 

5.9 

10 

25 

14.4 

13.5 

12.6 

11.7 

10.9 

10.0 

9.1 

8.3 

7.4 

6.5 

5.6 

5 

V     0 

14.6 

13.7 

12.7 

11.8 

10.9 

10.0 

9.1 

8.2 

7.3 

6.3 

5.4 

VII    0 

5 

14.7 

13.8 

12.8 

11.9 

10.9 

10.0 

9.1 

8.1 

7.2 

6.2 

5.3 

25 

10 

14.9 

13.9 

12.9 

12.0 

11.0 

10.0 

9.0 

8.0 

7.1 

6.1 

5.1 

20 

15 

15.0 

14.0 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

6.0 

5.0 

15 

20 

15.1 

14.1 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

5.9 

4.9 

10 

25 

15.1 

14.1 

13.1 

12.0 

11.0 

10.0 

9.0 

8.0 

6.9 

5.9 

4.9 

5 

VI    0 

15.1 

14.1 

13.1 

12.1 

11.0 

10.0 

9.0 

8.0 

6.9 

5.9 

4.9 

VI      0 

0" 

10" 

20' 

30" 

40" 

50" 

60" 

70" 

80" 

90'' 

100" 

92 


TABLE  LXXI. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Anomaly  corrected. 


0« 

diff. 

I* 

aiff. 

II* 

diff. 

III* 

diff. 

IV* 

iff. 

V* 

diff. 

o 

„ 

„ 

„ 

tt 

„ 

// 

o 

0 
1 
2 
3 
4 
5 

441.5 
441.5 
441.3 
441.1 
440.8 
440.4 

0.0 
0.1 
0.2 
0.3 
0.4 

404.1 
401.6 
399.2 
396.6 
394.0 
391.3 

2.5 

2.4  j 
2.6 
2.6 
2.7 

309.3 
305.6 
301.9 
298.1 
294.4 
290.6 

3.7 
3.7 
3.8 
3.7 
3.8  1 

195.3 
191.6 
187.9 
184.3 
180.6 
177.0 

3.7 
3.7 
3.6 
3.7 
3.6 

95.8 
93.0 
90.2 
87.6 
84.9 
82.3 

2.8 
2.8 
2.6 
2.7 
2.6 

30.6 
29.2 
27.8 
26.4 
25.1 
23.8 

1.4 
1.4 
1.4 
1.3 
1.3 

30 
29 

28 
27 
26 
25 

0.5 

2.7 

3.8 

3.6 

2.6 

1.2 

6 
7 
8 
9 
10 

439.9 
439.4 
438.7 
438.0 
437.2 

0.5 
0.7 
0.7 
0.8 

388.6 
385.8 
383.0 
380.1 
377.1 

2.8 
2.8 
2.9 
3.0 

286.8 
283.0 
279.2 
275.4 
271.5 

3.8 
3.8 
3.8 
3.9 

173.4 
169.8 
166.3 
162.8 
159.3 

3.6 
3.5 
3.5 
3.5 

79.7 
77.1 
74.6 
72.1 
69.7 

2.6 
2.5 
2.5 

2.4 

22.6 
21.4 
20.3 
19.2 
18.2 

1.2 
1.1 
1.1 
1.0 

24 
23 
22 
21 
20 

0.9 

3.0 

3.8 

3.5 

2.4 

1.0 

11 
12 
13 
14 
15 

436.3 
435.3 
434.2 
433.1 
431.8 

1.0 
1.1 
1.1 
1.3 

374.1 
371.1 
368.0 
364.8 
361.6 

3.0 
3.1 
3.2 
3.2 

267.7 
263.8 
260.0 
256.2 
252.3 

3.9 
3.8 
3.8 
3.9 

155.8 
152.4 
148.9 
145.5 
142.2 

3.4 
3.5 
3.4 
3.3 

67.3 
65.0 
62.7 
60.4 

58.2 

2.3 
2.3 
2,3 

2.2 

17.2 
16.3 
15.4 
14.6 
13.8 

0.9 
0.9 
0.8 
0.8 

19 
18 
17 
16 
15 

1.3 

3.2 

3.8 

3.3 

2.1 

0.7 

16 
17 

18 
19 
20 

430.5 
429.1 
427.6 
426.1 
424.5 

1.4 
1.5 
1.5 
1.6 

358.4 
355.1 
351.8 
348.4 
345.0 

3.3 
3.3 
3.4 
3.4 

248.5 
244.6 
240.8 
236.9 
233.1 

3.9 
3.8 
3.9 
3.8 

138.S 
135.6 
132.3 
129.1 
125.9 

3.3 
3.3 
3.2 
3.2 

56.1 
53.9 
51.9 
49.8 
47.9 

2.2 
2.0 
2.1 
1.9 

13.1 
12.4 
11.8 
11.2 
10.7 

0.7 

0.6 
0.6 
0.5 

14 
13 
12 
11 
10 

1.7 

3.4 

3.8 

3.2 

2.0 

0.5 

21 
22 
23 
24 
25 

422.7 
421.0 
419.1 
417.2 
415.2 

1.7 
1.9 
1.9 
2.0 

341.6 
338.1 
334.6 
331.1 
327.5 

3.5 
3.5 
3.5 
3.6 

229.3 
225.4 
221.6 
217.8 
214.0 

3.9 
3.8 
3.8 
3.8 

122.7 
119.6 
116.5 
113.4 
110.4 

3.1 
3.1 
3.1 
3.0 

45.9 
44.0 

42.2 
40.4 

38.7 

.9 

.8 
.8 
.7 

10.2 
9.8 
9.4 
9.1 

8,8 

0.4 
0.4 
0.3 
0.3 

9 
8 
7 
6 
5 

2.1 

3.5 

3.7 

3.0 

.7 

0.2 

26 
27 

28 
29 
30 

413.1 
410.9 
408.7 
406.4 
404.1 

2.2 
2.2 
2.3 
2.3 

324.0 
320.3 
316.7 
313.0 
309.3 

3.7 
3.6 
3.7 
3.7 

210.3 
206.5 
202.8 
199.0 
195.3 

3.8 
3.7 
3.8 
3.7 

107.4 
104.5 
i  101.6 
98.7 
95.8 

2.9 
2.9 
2.9 
2.9 

37.0 
35.3 
33.7 
32.1 
30.6 

.7 
.6 

.6 

• 

8.6 

8.4 
8.3 

8.2 
8.2 

0.2 
0.1 
0.1 
0.0 

4 
3 
2 
1 
0 

XI* 

X» 

IX* 

VIII* 

VII* 

VI* 

TABLE  LXXII.  93 

Moon's  Horary  Motion  in  Longitude. 
Arguments.   Sum  of  preceding  Equations,  and  Arg.  of  Variation. 


0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

8   ° 

0  0 

•> 

4.5 

5.5 

6.5 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.7 

14.7 

15.7 

16.7 

8   0 

XII  0 

5 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.6 

14.6 

15.6 

16.6 

25 

10 

4.8 

5.8 

6.8 

7.7 

8.7 

9.6 

10.6 

11.5 

12.5 

13.4 

14.4 

15.3 

16.3 

20 

15 

5.3 

6.1 

7.0 

7.9 

8.8 

9.7 

10.5 

11.4 

12.3 

13.1 

14.0 

14.9 

15.8 

15 

20 

5.8 

6.6 

7.4 

8.2 

8.9 

9.7 

10.5 

11.2ll2.0 

12.8 

13.5 

14.3 

15.1 

10 

25 

6.6 

7.2 

7.8 

8.5 

9.1 

9.7 

10.4 

ll.0|ll.7 

12.3 

12.9 

13.6 

14.2 

a 

I   0 

7.4 

7.8 

8.3 

8.8 

9.3 

9.8 

10.3 

10.8 

11.3 

11.8 

12.3 

12.7 

13.2 

XI  0 

5 

8.3 

8.6 

8.9 

9.2 

9.5 

9.9 

10.2 

10.5 

10.8 

11.2 

11.5 

11.8 

12.1 

25 

10 

9.2 

9.3 

9.5 

9.6 

9.8 

9.9 

10.1 

10.2 

10.4 

10.5 

10.7 

10.8 

11.0 

20 

15 

10.2 

10.1 

10.1 

10.1 

10.0 

10.0 

10.0 

10.0 

9.9 

9.9 

9.9 

9.8 

9.8 

15 

20 

11.1 

10.9 

10.7 

10.5 

10.3 

10.1 

9.9 

9.7 

9.5 

9-2 

9.0 

8.8 

8.6 

10 

25 

12.1 

11.7 

11.3 

10.9 

10.5 

10.2 

9.8 

9.4 

9.0 

8.6 

8.3 

7.9 

7.5 

5 

II  0 

12.9 

12.4 

11.8 

11.3 

10.8 

10.2 

9.7 

9.1 

8.6 

8.1 

7.5 

7.0 

6.4 

X  0 

5 

13.7 

13.0 

12.3 

11.6 

11.0 

10.3 

9.6 

8.9 

8.2 

7.5 

6.9 

6.2 

5.5 

25 

10 

14.3 

13.5 

12.7 

11.9 

11.1 

10.3 

9.5 

8.7 

7.9 

7.1 

6.3 

5.5 

4.7 

20 

15 

14.9 

14.0 

13.1 

12.2 

11.3 

10.4 

9.5 

8.6 

7.7 

6.8 

5.8 

4.9 

4.0 

15 

20 

15.3 

14.3 

13.3 

12.3 

11.4 

10.4 

9.4 

8.4 

7.5 

6.5 

5.5 

4.5 

3.6 

10 

25 

15.5 

14.5 

13.5 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4 

6.3 

5.3 

4.3 

3.3 

5 

III  0 

15.6 

14.5 

13.5 

12.5 

11.4 

10.4 

9.4 

8.4 

7.3 

6.3 

5.3 

4.2 

3.2 

IX  0 

5 

15.4 

14.4 

13.4 

12.4 

11.4 

10.4 

9.4 

8.4 

7.4 

6.4 

5.4 

4.4 

3.3 

25 

10 

15.2 

14.2 

13.3 

12.3 

11.3 

10.4 

9.4 

8.5 

7.5 

6.5 

5.6 

4.6 

3.6 

20 

15 

14.8 

13.9 

13.0 

12.1 

11.2 

10.4 

9.5 

8.6 

7.7 

6.8 

5.9 

5.1 

4.2 

15 

20 

14.2 

13.4 

12.6 

11.9 

11.1 

10.3 

9.5 

8.8 

8.0 

7.2 

6.4 

5.6 

4.9 

10 

25 

13.5 

12.9 

12.2 

11.6 

10.9 

10.3 

9.6 

9.0 

8.4 

7.6 

7.0 

6.3  5.7 

5 

IV  0 

12.7 

12.2 

11.7 

11.2 

10.7 

10.2 

9.7 

9.2 

8.7 

8.2 

7.7 

7.2  6.7 

VIII  0 

5 

11.9 

11.5 

11.2 

10.8 

10.5 

10.1 

9.8 

9.5 

9.1 

8.8 

8.4 

8.   7.7 

25 

10J10.9 

10.7 

10.6 

10.4 

10.2 

10.1 

9.9 

9.7 

9.6 

9.4 

9.2 

9.   8.9 

20 

15 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.  10.1 

15 

20 

8.9 

9.1 

9.3 

9.5 

9.7 

9.9 

10.1 

10.3 

10.5 

10.7 

10.9 

11. 

11.3 

10 

25 

8.0 

8.4 

8.7 

9.1 

9.5 

9.9 

10.2 

10.6 

11.0 

11.3 

11.7 

12. 

12.5 

5 

V  0 

7.1 

7.6 

8.2 

8.7 

9.2 

9.8 

10.3 

10.9 

11.4 

11.9 

12.5 

13.0 

13.6 

VII  0 

5 

6.3 

7.0 

7.6 

8.3 

9.0 

9.7 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

25 

10 

5.6 

6.4 

7.2 

8.0 

8.8 

9.7 

10.5 

11.3 

12.1 

13.0 

13.8 

14.6 

15.4 

20 

15 

5.0 

5.9 

6.8 

7.8 

8.7 

9.6 

10.6 

11.5 

12.4 

13.3 

14.3 

15.2 

16.1 

15 

20 

4.6 

5.6 

6.6 

7.6 

8.6 

9.6 

10.6 

11.6 

12.6 

13.6 

14.6 

15.7 

16.7 

10 

25 

4.3 

5.4 

6.4 

7.5 

8.5 

9.6  SlO.6  11.7 

12.7 

13.8 

14.9 

15.9 

17.0 

5 

VI  0 

4.2 

5.3 

6.4 

7.4 

8.5 

9.6  10.6  11.7 

12.8 

13.9 

14.9 

16.0 

17.1 

VI  0 

1 

ft 

" 

" 

0 

50 

100 

150 

200 

250  |  300  350 

400 

450 

500 

550 

600 

TABLE  LXXIII. 

Moon's  Horary  Motion  in  Longitude. 
Argument.     Argument  of  the  Variation. 


0* 

I* 

II* 

III* 

IV. 

V* 

0 

0 

77.2 

57.8 

20.3 

2.4 

21.5 

59.7 

o 
30 

1 

77.2 

56.7 

19.2 

2.5 

22.7 

60.9 

29 

2 

77.1 

55.5 

18.1 

2.6 

23.8 

62.0 

28 

3 

77.0 

54.3 

17.0 

2.7 

25.0 

63.1 

27 

4 

76.8 

53.1 

16.0 

2.9 

26.2 

64.2 

26 

5 

76.6 

51.8 

15.0 

3.1 

27.5 

65.3 

25 

6 

76.4 

50.5 

14.1 

3.3 

28.7 

66.3 

24 

7 

76.1 

49.3 

13.2 

3.7 

30.0 

67.3 

23 

8 

75.7 

48.0 

12.3 

4.0 

31.3 

68.3 

22 

9 

75.3 

46.7 

11.4 

4.4 

32.6 

69.2 

21 

10 

74.9 

45.4 

10.6 

4.9 

33.9 

70.1 

20 

11 

74.4 

44.1 

9.8 

5.3 

35.2 

70.9 

19 

12 

73.9 

42.8 

9.0 

5.9 

36.5 

71.7 

18 

13 

73.3 

41.5 

8.3 

6.4 

37.8 

72.5 

17 

14 

72.7 

40.2 

7.6 

7.0 

39.2 

73.3 

16 

15 

72.0 

38.9 

7.0 

7.7 

40.5 

74.0 

15 

16 

71.3 

37.5 

6.4 

8.3 

41.8 

74.7 

14 

17 

70.6 

36.2 

5.8 

9.1 

43.2 

75.3 

13 

18 

69.8 

34.9 

5.3 

9.8 

44.5 

75.8 

12 

19 

69.0 

33.6 

4.8 

10.6 

45.8 

76.4 

11 

20 

68.1 

32.3 

4.4 

11.5 

47.2 

76.9 

10 

21 

67.2 

31.1 

4.0 

12.3 

48.5 

77.3 

9 

22 

66.3 

29.8 

3.7 

13.2 

49.8 

77.7 

8 

23 

65.3 

28.6 

3.3 

14.2 

51.1 

78.1 

7 

24 

64.4 

27.3 

3.1 

15.1 

52.4 

78.4 

6 

25 

63.4 

26.1 

2.9 

16.1 

53.6 

78.6 

5 

26 

62.3 

24.9 

2.7 

17.1 

54.9 

78.  9 

4 

27 

61.2 

23.7 

2.5 

18.2 

56.1 

79.0 

3 

28 

60.1 

22.5 

2.5 

19.3 

57.3 

79.2 

2 

29 

59.0 

21.4 

2.4 

20.4 

58.5 

79.2 

1 

30 

57.8 

20.3 

2.4 

21.5 

59.7 

79.2 

0 

XI« 

X« 

IX* 

VIII* 

VII* 

VI* 

TABLE  LXXIV.  95 

Moon's  Horary  Motion  in  Longitude. 
Arguments.    Arg.  ot  Reduction  and  Sum  of  preceding  Equations 


0 

50  i  100 

150 

200 

250300 

350  400 

450 

500  550 

600 

650 

• 

0*  0 

3.3 

3.1  2.9 

2.7 

2.5 

2.3 

2.1 

.9 

.7 

1.5 

1.3 

.1 

0.9 

0.7 

XII  0 

5 

3.3 

3.1  J2.9  2.7 

2.5 

2.3 

2.1 

.911.7 

1.5 

1.3 

_1 

0.9 

0.7 

25 

10  |3.2 

3.0  2.8 

2.6 

2.4 

2.3 

2.1 

.9  .7 

1.5 

1.3 

.1 

1.0 

0.8 

20 

15  |3.1 

2.9 

2.8 

2.6 

2.4 

2.2 

2.1 

.9 

1.7 

1.5 

1.4 

.2 

1.0 

0.9 

15 

20 

3.0 

2.8 

2.7 

2.5 

2.4 

2.2 

2.1 

.9 

.8 

1.6 

1.5 

.3 

1.1 

1.0 

10 

25 

2.8 

2.7 

2.6 

2.4 

2.3 

2.2 

2.1 

1.9 

.8 

1.7 

1.5 

.4 

1.3 

1.2 

5 

I   0 

2.6 

2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

.8 

1.7 

1.6 

1.5 

1.4 

1.3 

XI   0 

5 

2.4 

2.4 

2.3 

2.2 

2.2 

2.J 

2.0 

2.0 

.9 

1.8 

1.8 

1.7 

1.6 

1.6 

25 

10 

2.2 

2.2 

2.2 

2.1 

2.1 

*2.0 

2.0 

2.0 

.9 

1.9 

1.9 

1.8 

1.8 

1.8 

20 

15 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

15 

20 

1.8 

.8 

1.8 

1.9 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

10 

25 

1.6 

.6 

1.7 

.8  il.8 

1.9 

2.0 

2.0 

2.1 

2.2 

2.2 

2.3 

2.4 

2.4 

5 

II  0 

1.4 

.5 

1.6 

.7  1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

X   0 

5 

1.2 

.3 

1.4 

.6  1.7 

1.8 

1.9 

2.1 

22 

2.3 

2.5 

2.6 

2.7 

2.8 

25 

10 

1.0 

.2 

1.3 

.5  1.6 

1.8 

1.9 

2.1 

2.2 

2.4 

2.5 

2.7 

2.9 

3.0 

20 

15 

0.9 

.1 

1.2 

.4 

1.6 

1.8 

1.9 

2.1 

2.3 

2.5 

2.6 

2.8 

3.0 

3.1 

15 

20 

0.8 

.0 

1.2 

1.4 

.6 

1  7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.0 

3.2 

10 

25 

0.7 

0.9 

1.1 

1.3 

c 

1  7 

1  9 

?  1 

9  3 

?,  5 

2.7 

2.9 

3.1 

3.3 

5 

III  0 

0.7 

0.9 

1.1 

1.3 

.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

IX   0 

5 

0.7 

0.9 

1.1 

1.3 

.5 

1.7 

1.9 

2. 

2.3 

2.5 

2.7 

2.9 

3.1 

3.3 

25 

10 

0.8 

.0 

1.2 

.4 

.6 

1.7 

1.9 

2. 

2.3 

2.5 

2.7 

2.9 

3.0 

3.2 

20 

15 

0.9 

.1 

1.2 

.4 

.6 

1.8 

1.9 

2. 

2.3 

2.5 

2.6 

2.8 

3.0  3.1 

15 

20 

1.0 

.2 

1.3 

.5 

.6 

1.8 

1.9 

2. 

2.2 

2.4 

2.5 

2.7 

2.9  3.0 

10 

25 

1.2 

.3 

1.4 

.6 

.7 

1.8 

1.9 

2. 

2.2 

2.3 

2.5 

2.6 

2.7  2.8 

5 

IV  0 

1.4 

.5 

1.6 

.7 

.8 

1.9 

2.0 

2. 

2.2 

2.3 

2.4 

2.5 

2.6  2.7 

VIII  0 

i 

5 

1.6 

1.6 

1.7 

.8 

.8 

1.9 

2.0 

2.0 

2.1 

22 

2.2 

2.3 

2.4  2.4 

25 

10 

1.8 

1.8 

1.8 

.9 

.9 

1.9 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2  2.2 

20 

15 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0  2.0 

15 

20 

2.2 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

1.9 

1.9 

1.9 

.8 

.8  1.8 

10 

25 

2.4 

2.4 

2.3 

2.2 

2.2 

2.1 

2.0 

2.0 

1.9 

1.8 

1.8 

.7 

.e  1.6 

5 

V  0 

2.6 

2.5 

2.4 

2.3 

2.2 

2.1 

2.0 

1.9 

1.8 

1.7 

1.6 

.5 

.4 

1.3 

Vll  0 

5 

2.8 

2.7 

2.6 

2.4 

2.3 

22 

2. 

1.9 

1.8 

1.7 

1.5 

.4 

.3 

1.2 

25 

10 

3.0 

2.8 

2.7 

2.5 

2.4 

2.2 

j 

1.9 

1.8 

1.6 

1.5 

.3 

.1 

1.0 

20 

15 

3.1 

2.9 

2.8 

2.6 

2.4 

2.2 

2. 

1.9 

1.7 

1.5 

1.4 

.2 

.0 

0.9 

15 

20 

3.2 

3.0 

2.8 

2.6 

2.4 

2.3 

2. 

1.9 

1.7 

1.5 

1.3 

.1 

1.0 

0.8 

10 

25 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

2. 

1.9 

1.7 

1.5 

1.3 

.1 

0.9 

0.7 

5 

VI  0 

3.3 

3.1 

2.9 

2.7 

2.5 

2.3 

j 

1.9 

1.7 

1.5 

1.3 

.1 

0.9 

0.7 

VI   0 

0 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

550 

600 

650 

96        TABLE  LXXV. 
Moon's  Horary  Motion  in  Long. 
Arg.    Arg.  of  Reduction. 


TABLE  LXXVI. 
Moon's  Horary  Motion  in  Long. 

(Equation  of  the  second  order.) 
Arguments.    Arg's.of  Table  LXX 


Os  Via 

Is  Vis  Us  Villa 

i 

o 

„ 

„ 

>t 

0 

0 

2.1 

6.0 

14.0 

30 

1 

2.1 

6.3 

14.2 

29 

2 

2.1 

6.5 

14.4 

28 

3 

2.1 

6.8 

14.7 

27 

4 

2.2 

7.0 

14.9 

26 

5 

2.2 

7.3 

15.1 

25 

6 

2.2 

7.5 

15.3 

24 

7 

2.3 

7.8 

15.5 

23 

8 

2.4 

8.1 

15.7 

22 

9 

2.5 

8.4 

15.9 

21 

10 

2.5 

8.6 

16.1 

20 

11 

2.6 

8.9 

16.2 

19 

12 

2.7 

9.2 

16.4 

18 

13 

2.9 

9.4 

16.6 

17 

14 

3.0 

9.7 

16.7 

16 

15 

3.1 

10.0 

16.9 

15 

16 

3.3 

10.3 

17.0 

14 

17 

3.4 

10.6 

17.1 

13 

18 

3.6 

10.8 

17.3 

12 

19 

3.8 

11.1 

17.4 

11 

20 

3.9 

11.4 

17.5 

10 

21 

4.1 

11.6 

17.5 

9 

22 

4.3 

11.9 

17.6 

8 

23 

4.5 

12.2 

17.7 

7 

54 

4.7 

12.5 

17.8 

6 

25 

4.9 

12.7 

17.8 

5 

26 

5.1 

13.0 

17.8 

4 

27 

5.3 

13.2 

17.9 

3 

28 

5.6 

13.5 

17.9 

2 

29 

5.8 

13.7 

17.9 

1 

30 

6.0 

14.0 

17.9 

0 

XTs  Va 

XslVs 

IXs  Ilk 

Arg. 

0 

50 

„ 

100      ' 

1" 

f              0 

/, 

ri 

0        0 

0.05 

0.05 

0.05 

I         0 

0.08 

0.05 

0.02 

II        0 

0.10 

0.05 

0.00 

III      0 

0.10 

0.05 

0.00 

IV      0 

0.09 

0.05 

0.01 

V        0 

0.07 

0.05 

0.03 

VI      0 

0.05 

0.05 

0.05 

VII     0 

0.03 

0.05 

0.07 

VIII   0 

0.01 

0.05 

0.09 

IX      0 

0.00 

0.05 

0.10 

X       0 

0.00 

0.05 

0.10 

XI      0 

0.02 

0.05 

0.08 

XII    0 

0.05 

0.05 

0.05 

0 

ft 

50 

100 

Constant  to  be  added  27'24".0. 

TABLE  LXXVII. 
Moon's  Horary  Motion  in  Longitude. 

(Equations  of  the  second  order.) 
Arguments.     Arguments  of  Tables  LXXII  and  LXXIV. 


Variation. 

Reduction. 

0 

600 

100 

200 

300 

400 

500 

600 

0 

•          «      o 
0.     VI.      0 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

0.14 

003 

0.03 

I.      VII.    0 

0.22 

0.19 

0.16 

0.13 

0.10 

0.06 

0.02 

0.01 

0.05 

I.       VII.  15 

0.23 

0.20 

0.17 

0.13 

0.10 

005 

0.01 

0.01 

0.06 

II.     VIII.  0 

0.22 

0.19 

0.16 

0.1.3 

0.10 

0.07 

0.03 

0.01 

0.05 

III.    IX.      0 

0.14 

0.14 

0.14 

0.14 

0.14 

O.H 

0.14 

0.03 

0.03 

IV.    X.       0 

0.06 

0.09 

0.12 

0.15 

0.18 

0.21 

0.26 

0.05 

0.01 

IV.    X.     15 

0.05 

0.08 

0.11 

0.15 

0.18 

0.23 

0.28 

0.05 

0.00 

V.     XI.      0 

0.06 

0.09 

0.12 

0.15 

0.18 

0.22 

0.26 

0.05 

0.01 

VI.    XII.    0 

0.14 

0.14 

0.14  |0.14 

0.14 

C.14 

0.14 

0.03 

0.03 

TABLE  LXXVIII.  97 

Moon's  Horary  Motion  in  Longitude 

(Equations  of  the  second  order.) 
Arguments.     Args.  of  Evection,  Anomaly,  Variation,  Reduction. 


Evec. 

Anom. 

Var. 

Red. 

Evec. 

Anom. 

Var. 

Red. 

«        ° 
0         0 

0.16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

XII       0 

5 

0.15 

0.93 

0.28 

0.09 

0.18 

1.17 

0.40 

0.06 

25  . 

10 

0.13 

0.81 

0.22 

0.10 

0.19 

1.28 

0.46 

0.05 

20 

15 

0.12 

0.70 

0.17 

0.11 

0.21 

1.40 

0.51 

0.04 

15  i 

20 

0.10 

0.59 

0.12 

0.12 

0.22 

1.50 

0.56 

0.03 

10 

25 

0.09 

0.49 

0.08 

0.13 

0.24 

1.60 

0.60 

0.02 

5 

I           0 

0.08 

0.40 

0.05 

0.14 

0.25 

1.70 

0.63 

0.01 

XI        0 

5 

0.07 

0.31 

0.02 

0.15 

0.26 

1.78 

0.66 

0.01 

25 

10 

0.05 

0.24 

0.01 

0.15 

0.27 

1.86 

0.67 

0.00 

20 

15 

0.04 

0.17 

0.01 

0.15 

0.28 

1.92 

0.67 

0.00 

15 

20 

0.03 

0.12 

0.01 

0.15 

0.29 

1.98 

0.67 

0.00 

10 

25 

0.03 

0.07 

0.03 

0.15 

0.30 

2.02 

0.65 

0.01 

5 

n      o 

0.02 

0.04 

0.06 

0.14 

0.31 

2.05 

0.62 

0.01 

X          0 

5 

0.01 

0.02 

0.09 

0.13 

0.32 

2.08 

0.59 

0.02 

25 

10 

0.01 

0.00 

0.13 

0.12 

0.32 

2.09 

0.54 

0.03 

20 

15 

0.00 

0.00 

0.18 

0.11 

0.32 

2.10 

0.50 

0.04 

15 

20 

0.00 

0.00 

0.24 

0.10 

0.33 

2.09 

0.44 

0.05 

10 

25 

0.00 

0.02 

0.29 

0.09 

0.33 

2.08 

0.39 

0.06 

5 

III        0 

0.00 

0.04 

0.35 

0.08 

0.33 

2.06 

0.33 

0.08 

IX         0 

5 

0.00 

0.07 

0.40 

0.06 

0.33 

2.03 

0.27 

0.09 

25 

10 

0.01 

0.10 

0.46 

0.05 

032 

2.00 

0.22 

0.10 

20 

15 

0.01 

0.14 

0.51 

0.04 

0.32 

1.96 

0.17 

0.11 

15 

20 

o-oi 

0.18 

0.56 

0.03 

0.31 

1.91 

0.12 

0.12 

10 

25 

0.02 

'  0.23 

0.60 

0.02 

0.31 

1.87 

0.08 

0.13 

5 

IV         0 

0.03 

0.28 

0.63 

0.01 

0.30 

1.82 

0.05 

0.14 

VIII     0 

5 

0.03 

0.34 

0.66 

0.01 

0.29 

1.76 

0.02 

0.15 

25 

10 

0.04 

0.39 

0.67 

0.00 

0.28 

1.70 

0.01 

0.15 

20 

15 

0.05 

0.45 

0.68 

0.00 

0.27 

1.64 

0.00 

0.15 

15 

20 

0.06 

0.52 

0.67 

0.00 

0.26 

1.58 

0.00 

0.15 

10 

25 

0.08 

•0.58 

0.66 

0.01 

0.25 

1.52 

0.02 

0.15 

5 

V          0 

0.09 

0.64 

0.64 

0.01 

0.24 

1.45 

0.04 

0.14 

VII       0 

5 

0.10 

0.71 

0.60 

0.02 

0.23 

1.39 

0.08 

0.13 

25 

10 

0.11 

0.78 

0.56 

0.03 

0.22 

1.32 

0.12 

0.12 

20 

15 

0.12 

0.84 

0.51 

0.04 

0.20 

1.25 

0.16 

0.11 

hi 

SO 

0.14 

0.91 

0.46 

0.05 

0.19 

1.18 

0.22 

0.10 

10 

25 

0.15 

0.98 

0.40 

0.06 

0.18 

1.12 

0.28 

0.09 

5 

VI        0 

0.16 

1.05 

0.34 

0.08 

0.16 

1.05 

0.34 

0.08 

VI         0 

98 


TABLE  LXXIX. 

Moon's  Horary  Motion  in  Latitude. 
Argument.     Arg.  I  of  Latitude. 


0 

I* 

II* 

III* 

IV* 

V* 

o 
0 

378.0 

354.3 

289.2 

200.0 

110.8 

45.7 

o 
30 

1 

378.0 

352.7 

286.5 

196.9 

108.1 

44.2 

29 

2 

377.9 

351.1 

283.8 

193.8 

105.4 

42.7 

28 

3 

3778 

349.4 

281.0 

190.7 

102.8 

41.3 

27 

4 

377.6 

347.7 

278.3 

187.5 

100.2 

39.9 

26 

5 

377.3 

346.0 

275.5 

184.4 

97.7 

38.6 

25 

6 

377.0 

344.2 

272.6 

181.3 

95.1 

37.3 

24 

7 

376.7 

342.3 

269.8 

178.2 

92.6 

36.1 

23 

8 

376.3 

340.5 

266.9 

175.1 

90.2 

34.9 

22 

9 

375.8 

338.5 

264.0 

172.1 

87.7 

33.8 

21 

10 

375.3 

336.6 

261.1 

169.0 

85.3 

32.7 

20 

11 

374.7 

334.5 

258.1 

165.9 

83.0 

31.6 

19 

12 

374.1 

332.5 

255.2 

162.9 

80.7 

30.7 

18 

13 

373.5 

330.4 

252.2 

159.8 

78.1 

29.7 

17 

14 

372.7 

328.3 

249.2 

156.8 

76.1 

28.9 

16 

15 

372.0 

326.1 

246.2 

153.8 

73.9 

28.0 

15 

16 

371.1 

323.9 

243.2 

150.8 

71.7 

27.3 

14 

17 

370.3 

321.9 

240.2 

147.8 

69.6 

26.5 

13 

18 

369.3 

319.3 

237.1 

144.8 

67.5 

25.9 

12 

19 

368.4 

317.0 

234.1 

141.9 

65.5 

25.3 

11 

20 

367.3 

314.7 

231.0 

138.9 

63.4 

24.7 

10 

21 

366.2 

312.3 

227.9 

136.0 

61.5 

24.2 

9 

22 

365.1 

309.8 

224.9 

133.1 

59.5 

23.7 

8 

23 

363.9 

307.4 

221.8 

130.2 

57.7 

23.3 

7 

24 

362.7 

304.9 

218.7 

127.4 

55.8 

23.0 

6 

25 

361.4 

302.3 

215.6 

124.5 

54.0 

22.7 

5 

26 

360.1 

299.8 

212.5 

121.7 

52.3 

22.4 

4 

27 

358.7 

297.2 

209.3 

119.0 

50.6 

22.2 

3 

28 

357.3 

294.6 

206.2 

116.2 

48.9 

22.1 

2 

29 

355.8 

291.9 

203.1 

113.5 

47.3 

22.0 

1 

30 

354.3 

289.2 

200.0 

110.8 

45.7 

22.0 

0 

XI* 

X* 

IX* 

VIII* 

VII* 

VI* 

• 

TABLE  LXXX. 

Moon's  Horary  Motion  in  Latitude. 
Arguments.  Args.  V,  VI,  VII,  VIII,  IX,  X,  XI,  and  XII,  of  Latitude. 


Arg. 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

Arg. 

0 

1000 

0.00 

0.50 

0.34 

0.00 

0.50 

0.04 

0.12 

0.08 

50  0.01 

0.49 

0.33 

0.00 

0.49 

0.04 

0.12 

0.07 

950 

100 

0.04 

0.45 

0.30 

0.02 

0.45 

0.04 

0.11 

0.05 

900 

150 

0.09 

0.40 

0.27 

0.04 

0.40 

0.03 

0.10 

0.03 

850 

200 

0.16 

0.33 

0.22 

0.06 

0.33 

0.03 

0.08 

0.01 

800 

250 

0.23 

0.25 

0.17 

0.09 

0.25 

0.02 

0.06  0.00 

750 

300 

0.30 

017 

0.12 

0.12 

0.17 

0.01 

0.040.01 

700 

350 

0.37 

0.10 

0.07 

0.14 

0.10 

0.01 

0.020.03 

650 

400 

0.42 

0.05 

0.04 

0.16 

0.05 

0.00 

0.01,0.05 

600 

450 

0.45 

0.01 

0.01 

0.18 

0.01 

0.00 

0.00  0.07 

550 

500 

0.46 

0.00  0.00  0.18  0  00  0.00 

0.00  0.08 

500 

TABLE  LXXXI.     Moon's  Horary  Motion  in  Latitude.        99 

Arguments.     Preceding  equation,   and  Sum  of  equations   of  Horary 

Motion  in  Longitude,  except  the  last  two. 


T> 

0" 

50" 

100" 

150"  1200"  250" 

300" 

350' 

400" 

4JbQ" 

500' 

550" 

600" 

650" 

I 

rl. 
eq. 

1".6 

1".4 

l."l 

0".9 

0".6 

0".4 

0".l 

0''.2 

0".4 

0".7 

0".9 

1".2 

1".4 

1".7 

Dtf. 

20  59.0 

54.5 

50.0 

45.4 

40.9 

36.4 

31.8 

27.3 

22.8 

18.2 

13.7 

9.1 

4.6 

0.1 

4.5 

3057.453.1 

48.9 

44.6 

40.3 

36.0 

31.7 

27.4 

23.2 

18.9 

14.6 

10.3 

6.0 

1.7 

4.3 

4055.851.8 

47.7 

43.7 

39.7 

35.6 

31.6 

27.6 

23.6 

19.5 

15.5 

11.5 

7.4 

3.i 

4.0 

50  54.2  50.4 

46.6 

42.9 

39.1 

35.3 

31.5 

27.7 

24.0 

20.2 

16.4 

12.6 

8.8     5.1 

3.8 

6052.649.1 

45.5 

42.0 

38.5 

34.9 

31.4 

27.9 

24.4 

20.8 

17.3 

13.8 

10.2     6.7 

3.5 

7051.047.7 

44.4 

41.1 

37.9 

34.6 

31.3 

28.0 

24.8 

21.5 

18.2 

14.9 

11.7 

8.4 

3.3 

8049.346.3 

43.3 

40.3 

37.3 

34.2 

31.2 

28.2 

25.2 

22.1 

19.1 

16.1 

13.1 

10.0 

3.0 

9047.745.0 

42.2 

39.4 

36.7 

339 

31.1 

28.3 

25.6 

22.8 

20.0 

17.3 

14.5 

11.7 

2.8 

10046.1 

43.6 

41.1 

38.6 

36.0 

33.5 

31.0 

28.5 

26.0 

23.4 

20.9 

18.4 

15.9  13.4 

2.5> 

11044.5 

42.2 

40.0 

37.7 

35.4 

33.2 

30.9 

28.6 

26.4 

24.1 

21.8 

19.6 

17.3  15.0 

2.3 

120  42.9  40.9 

38.9 

36.9 

34.8 

32.8 

30.8 

28.8 

26.8 

24.8 

22.7 

20.7 

18.7   16.7 

2.0 

13041.339.5 

37.8 

36.0 

34.2 

32.5 

30.7 

28.9 

27.2 

25.4 

23.7 

21.9 

20.1J  18.4 

1.8 

140 

39.7  38.2 

36.7 

35.1 

33.6 

32.1 

30.6 

29.1 

27.6 

26.1 

24.6 

23.0 

21.5  20.0 

1.5 

150 

38.  1  36.8 

35.5 

34.3 

33.0 

31.8 

30.5 

29.2 

28.0 

26.7 

25.5 

24.2 

23.0  21.7 

1.3 

160 

36.5 

35.4 

34.4 

33.4 

32.4 

31.4 

30.4 

29.4 

28.4 

27.4 

26.4 

25.4 

24.4  23.3 

1.0 

170 

34.8 

34.1 

33.3 

32.6 

31.8 

31.1 

30.3 

29.5 

28.8 

28.0 

27.3 

26.5 

25.8  25.0 

0.8 

180 

33.2 

32.7 

32.2 

31.7 

31.2 

30.7 

30.2 

29.7 

29.2 

28.7 

28.2 

27.7 

27.2  26.7 

0.5 

190 

31.6 

31.4 

31.1 

30.9 

30.6 

30.4 

30.1 

29.8 

29.6 

29.3 

29.1 

28.8 

28.6  28.3 

0.3 

200 

30.0 

30.0 

30.0 

30.0|  30.0 

30.0 

30.0 

30.0 

30.0 

80.0 

30.0 

30.0 

30.0  30.0 

0.0 

210 

28.4 

28.6 

28.9 

29.1 

29.4 

29.6 

29.9 

30.2 

30.4 

30.7 

30.9 

31.2 

31.4'  31.7 

0.3 

220 

26.8  27.3  27.8 

28.3  28.8 

29.3 

29.8  30.3 

30.8 

31.3 

31.8 

32.3 

32.8!  33.3 

0.5 

230 

25.225.9   26.7 

27.4  28.2 

28.9 

29.7 

30.5 

31.2 

32.0 

32.7 

33.5 

34.2  35.0 

0.8 

240  23.5  24.6  25.6 

26.6  27.6 

28.6 

29.6 

30.6 

31.6 

32.6 

33.6 

34.6 

35.6  36.7 

1.0 

25021.923.2  24.5 

25.7  27.0 

28.2 

29.5 

30.8 

32.0 

33.3 

34.5 

35.8 

37.1  38.3 

1.3 

260,20.3  21.8  23.3 

24.9 

26.4 

27.9 

29.4 

30.9 

32.4 

33.9 

35.4 

37.0 

38.5 

40.0 

1.5 

270 

18.720.5  22.2 

24.0 

25.8 

27.5 

29.3 

31.1 

32.8 

34.6 

36.3 

38.1 

39.9 

41.6 

1.8 

280 

17.1 

19.1  21.1 

23.1 

25.2 

27.2 

29.2 

31.2 

33.2 

35.2 

37.3 

39.3 

41.3  43.3 

2.0 

290 

15.5 

17.8  20.0  22.3 

24.6 

26.8 

29.1 

31.4 

33.6 

35.9 

38.2 

40.4 

42.7 

45.0 

2.3 

300 

13.9 

16.4  18.9  21.4 

24.0 

26.5 

29.0 

31.5 

34.0 

36.6 

39.1 

41.6 

44.1 

46.6 

2.5 

310  12.3 

15.0   17.8 

20.6 

23.3 

26.1 

28.9 

31.7 

34.4 

37.2 

40.0 

42.7 

45.5 

48.3 

2.8 

320 

10.7 

13.7 

16.7  19.7 

22.7 

25.8 

28.8 

31.8 

34.8 

37.9 

40.9 

43.9 

46.9 

50.0 

3.0 

330 

9.0 

12.3 

15.6J  18.9 

22.1 

25.4 

28.7 

32.0 

35.2 

38.5 

41.8 

45.1 

4S.3J  51.6 

33 

340 

7.4 

10.9 

14.5  18.0 

21.5 

25.1 

28.6  32.1  35.6 

39.2 

42.7 

46.2 

49.8!  53.3  35 

350    5.8 

9.6 

13.4 

17.1 

20.9 

24.7 

28.5  32.3  36.0 

39.8 

43.6 

47.4 

51.2  54.9 

3.8 

360    4.2 

8.2 

12.3 

16.3 

20.3 

24.4 

23.4  32.4  36.4 

40.5 

44.5 

48.5 

52.6 

56.6  4.0 

370    2.6 

6.9 

11.1 

15.4 

19.7 

24.0 

28.3  32.6 

36.8 

41.1 

45.4  49.7 

54.0 

58.3  43 

380 

1.0 

5.5 

10.0 

14.6 

19.1 

23.6 

28.2  32.7 

37.2 

41.8 

46.3  50.9 

55.4 

59.9  4.5 

i 

0" 

50" 

100"!  150" 

200" 

250" 

300"  350" 

400" 

450" 

500"!550" 

600" 

650^1 

TABLE    LXXXII.     Moon's  Horary  Motion  in  Latitude. 
Argument.     Arg    II.  of  Latitude. 


O* 

I* 

Us 

III* 

IV* 

V* 

0 

0 

9.3 

8.7 

7.1 

5.0 

2.9 

1.3 

0 

30 

3 

9.3 

8.6 

6.9 

4.8 

2.7 

1.2 

27 

6 

9.2 

8.5 

6.7 

4.6 

2.5 

1.1 

24 

9 

9.2 

8.3 

6.5 

4.3 

2.3 

1.0 

21 

12 

9.2 

8.2 

6.3 

4.1 

2.1 

0.9 

18 

15 

9.1 

8.0 

6.1 

3.9 

2.0 

0.9 

15 

18 

9.1 

7.9 

5.9 

3.7 

1.8 

0.8 

12 

21 

9.0 

7.7 

5.7 

3.5 

1.7 

0.8 

9 

24 

8.9 

7.5 

5.4 

3.3 

1.5 

0.8 

6 

27 

88 

7.3 

5.2 

3.1 

1.4 

0.7 

3 

30 

S7 

7.1 

5.0 

2.9 

1.3 

0.7 

0 

XI* 

X*      IX.N  IVIII* 

vn* 

VI* 

100  TABLE  LXXXIII. 

Moon's  Horary  Motion  in  Latitude. 

Arguments.  Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  the  last  two. 


TABLE  LXXXIV. 

Moon's  Hor.  Motion  in  La\ 

(Equa.  of  second  order.) 

Argument.    Arg.  I  of  Lat. 


Free. 

equ. 

0 

100 

200 

300 

400 

500 

600 

n 
700 

0 

2.1 

1.8 

1.5 

1.2 

0.9 

0.6 

0.3 

0.0 

1 

1  9 

1.6 

1.4 

.1 

0.9 

0.7 

0.4 

0.2 

2 

1.7 

1.5 

.3 

[ 

.0 

0.8 

0.6 

0.3 

3 

1  5 

1  4 

.2 

.0 

0.9 

0.8 

0.6 

4 

1.3 

1.2 

.2 

, 

.1 

1.0 

0.9 

0.9 

5 

1.1 

1.1 

.1 

. 

.1 

1.1 

1.1 

1.1 

6 

09 

1.0 

.0 

.1 

1.2 

1.3 

1.3 

7 

0.7 

0.8 

.0 

.2 

1.3 

1.4 

1.6 

8 

05 

0.7 

0.9 

.2 

1.4 

1.6 

1.9 

9 

03 

0.6 

0.8 

1.1 

.3 

1.5 

1.8 

2.0 

10 

0.1 

0.4 

0.7 

1.0 

.3 

1.6 

1.9 

2.2 

500 

600 

700 

0 

100 

200 

300 

400 

Constant  to  be  subtracted  237"  .2. 

TABLE  LXXXV. 

Moon's  Horary  Motion  in  Latitude. 

(Equations  of  second   order.) 
Arguments.     Preceding  equation,  and  Sum 
of  equations  of  Horary  Motion  in  Longi- 
tude, except  the  last  two. 


Free. 

equ. 

0 

100 

200 

300 

400 

500 

600 

700 

0.00 

0.65 

0.57 

0.48 

0.39 

0.31 

0.21 

0.12 

0.00 

0.10  0.62 

055 

0.47 

0.39 

0.31 

0.23 

0.15 

0.04 

0.20  0.69 

0.53 

0.46 

0.39 

0.32 

0.25 

0.18 

0.09 

0.30  0.66 

0.51 

0.45 

0.39 

0.33 

0.27 

0.21 

0.13 

0.40  0.63 

0.48 

0.44 

0.39 

0.34 

0.29 

0.24 

0.17 

0.50  0.50 

0.46 

0.43 

0.38 

0.35 

0.30 

0.27 

0.21 

0.60  '0.47 

0.44 

0.42 

0.38 

0.36 

0.32 

0.29 

0.25 

0.70  0.44 

0.42 

0.40 

0.38 

0.36 

0.34 

0.32 

0.30 

0.80 

0.41 

0.40 

0.39 

0.38 

0.37 

0.36 

0.35 

0.34 

0.90 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

0.38 

1.00 

0.35 

0.36 

0.37 

0.38 

0.39 

0.40 

0.41 

0.42 

1.10 

0.32 

0.34 

0.36 

0.38 

0.40 

0.42 

0.44 

0.46 

1.20 

0.29 

0.32 

0.34 

0.38 

0.40 

0.44 

0.47 

0.51 

1.30 

0.26 

0.30 

033 

0.38 

0.41 

0.46 

0.49 

0.55 

1.40 

0.23 

0.28 

0.32 

0.37 

0.42 

0.47 

0.52 

0.59 

1.50 

0.20 

0.25 

0.31 

0.37 

0.43 

0.49 

0.55 

0.63 

1.60 

0.17 

0.23 

0.30 

0.37 

0.44 

0.51 

0.58 

0.67 

1.70 

0.14 

0.21 

0.29 

0.37 

0.45 

0.53 

0.61 

0.72 

1.80 

0.11 

0.19 

0.28 

0.37 

0.45 

0.55 

0.64 

0.76 

0 

100 

200 

300 

400 

500 

600 

700 

I 

I 

0  0 

// 

// 

•  ° 

0  0 

0.90 

0.90 

XII  0 

5 

0.83 

0.97 

25 

10 

0.75 

1.05 

20 

15 

0.68 

.12 

15 

20 

0.61 

.19 

10 

25 

0.54 

.26 

5 

I   0 

0.47 

.33 

XI   0 

5 

0.41 

.39 

25 

10 

0.35 

.45 

20 

15 

0.29 

.51 

15 

20 

0.24 

.56 

10 

25 

0.20 

.60 

5 

II  0 

0.16 

.64 

X   0 

5 

0.12 

.68 

25 

10 

0.09 

.71 

20 

15 

0.07 

.73 

15 

20 

0.05 

1.75 

10 

25 

0.04 

1.76 

5 

III  0 

0.04 

1.76 

IX   0 

5 

0.04 

.76 

25 

10 

0.05 

.75 

20 

15 

0.07 

.73 

15 

20 

0.09 

.71 

10 

25 

0.12 

.68 

5 

IV  0 

0.16 

.64 

VIII  0 

5 

0.20 

.60 

25 

10  0.24 

.56 

20 

15  0.29 

.51 

15 

20  0.35 

.45 

10 

25  !0.41 

.39 

5 

V  0 

0.47 

1.33 

VII   0 

5 

0.54 

1.26 

25 

10  ,0.61 

1.19 

20 

15  0.68 

1.12 

15 

20  0.75 

1.05 

10 

25  0.83 

0.97 

5 

VI  0  0.90 

0.90 

VI   0 

TABLE  LXXXVI. 
Mean   New   Moons   and  Arguments,   in   January. 


101 


Years. 

Mean  New 
Moon  in 
January. 

L 

n. 

m. 

IV. 

N. 

1836  B 

d.  h.  m. 
17  10  32 

0469 

1246 

17 

08 

669 

1837 

5  19  20 

0171 

9852 

00 

97 

692 

1838 

24  16  53 

0681 

9175 

99 

85 

799 

1839 

14  1  42 

0383 

7780 

82 

74 

822 

1840  B 

3  10  30 

0085 

6386 

65 

63 

844 

1841 

21  8  3 

0595 

5709 

63 

51 

951 

1842 

10  16  51 

0297 

4314 

46 

40 

974 

1843 

29  14  24 

0807 

3637 

44 

28 

081 

1844  B 

18  23  13 

0509 

2243 

28 

17 

104 

1845 

7  8  1 

0211 

0848 

11 

06 

126 

1846 

26  5  34 

0721 

0171 

09 

94 

234 

1847 

15  14  22 

0423 

8777 

92 

84 

256 

1848  B 

4  23  Jl 

0125 

7382 

75 

73 

278 

1849 

22  20  43 

0635 

6705 

73 

61 

386 

1850 

12  5  32 

0337 

6311 

56 

50 

408 

1851 

1  14  21 

0038 

3916 

40 

39 

431 

1852  B 

20  11  53 

0549 

3239 

38 

27 

538 

1853 

8  20  42 

0251 

1845 

21 

16 

560 

1854 

27  18  14 

0761 

1168 

19 

04 

668 

1855 

17  3  3 

0463 

9773 

02 

93 

690 

1856  B 

6  11  51 

0164 

8379 

85 

82 

713 

1857 

24  9  24 

0675 

7702 

84 

70 

820 

1858 

13  18  13 

0376 

6307 

67 

59 

843 

1859 

3  3  1 

0078 

4913 

50 

48 

865 

1860  B 

22  0  34 

0588 

4236 

48 

36 

972 

1861 

10  9  23 

290 

2842 

31 

25 

994 

1862 

29  6  55 

800 

2164 

29 

13 

102 

1863 

18  15  44 

502 

770 

12 

2 

124 

1864B 

8  0  32 

204 

9376 

95 

91 

146 

1865 

25  22  5 

714 

8699 

94 

79 

254 

1866 

15  6  54 

416 

7304 

77 

68 

276 

1867 

4  15  42 

117 

5910 

60 

57 

299 

1868  B 

23  13  15 

628 

5234 

58 

46 

406 

1869 

11  22  3 

329 

3838 

41 

35 

428 

1870 

30  19  36 

840 

3161 

40 

23 

536 

1871 

20  4  25 

541 

1767 

23 

12 

558 

1872  B 

9  13  13 

243 

372 

6 

1 

581 

1873 

27  10  46 

753 

9695 

4 

89 

688 

1874 

16  19  34 

455 

8301 

87 

78 

710 

1875 

6  4  23 

157 

6906 

70 

67 

733 

1876  B 

25  1  55 

667 

6229 

69 

55 

840 

1877 

13  10  44 

369 

4835 

52 

44 

862 

1878 

2  19  33 

71 

3441 

35 

33 

885 

1879 

21  17  5 

581 

2763 

33 

21 

993 

1880  B 

11  1  54 

283 

1369 

16 

10 

15 

1881 

28  23  27 

793 

692 

14 

99 

123 

1882 

18  8  15 

495 

9297 

98 

88 

145 

1883 

7  17  4 

197 

7903 

81 

77 

167 

1884  B 

26  14  36 

707 

7226 

79 

65 

275 

1885 

14  23  25 

409 

5832 

62 

54 

297 

35 


1.02 


TABLE  LXXX\  II 


Mean  Lunations  and  Changes  of  the  Arguments. 


Num 

Lunations. 

I. 

II. 

III. 

IV. 

N. 

(I.  h  m 

4 

14  18  22 

404 

5359 

58 

50 

43 

29  12  44 

808 

717 

15 

99 

85 

2 

59  1  28 

1617 

1434 

31 

98 

170 

3 

88  14  12 

2425  ,  2151 

46 

97 

256 

.  4 

118  2  56 

3234 

2869 

61 

96 

341 

5 

147  15  40 

4042 

3586 

76 

95 

426 

6 

177  4  24 

4851 

4303 

92 

95 

511 

7 

206  17  8  ' 

5659 

5020 

7 

94 

596 

8 

236  5  52 

6468 

5737 

22 

93 

682 

9 

265  18  36 

7276 

6454 

37 

92 

767 

10 

295  7  20 

8085 

7171 

53 

91 

852 

11 

324  20  5 

8893 

7889 

68 

90 

937 

12 

354  8  49 

9702 

8606 

83 

89 

22 

13 

383  21  33 

510 

9323 

98 

88 

108 

TABLE  LXXXVIII. 


Number  of  Days  from    the  commencement  of  the  vear 
to  the  first  of  each  month* 


Months. 

Com. 

Bis. 

January 

Days. 
0 

Days. 
0 

February 

31 

31 

March 

59 

60 

April 

90 

91 

May 

120 

121 

June 

151 

152 

JuJy 

181 

182 

August 

212 

213 

September 
October     . 

243 
273 

244 
274 

November 

304 

305 

December 

334 

335 

TABLE 


Equations  for  Neiv  and  Full  Moon. 


** 

I  J  II 

Arg. 

I 

II 

Arg  HIj  IV 

Arg 

h  m   h  rti 

h  m 

h  m 

m 

771 

0 

4  20  10  10 

5000 

4  20 

10  10 

25 

3 

31 

26 

160  4  36  '  9  36 

5100 

4  5 

10  50 

26 

3 

31 

24 

200  4  52!  9  2 

5200 

3  49 

11  30 

27 

3 

30 

23 

300  5  8   8  28 

5300 

3  34 

12  9 

28 

3 

30 

22 

400  5  24  7  55 

5400 

3  19 

12  48 

29 

3 

30 

21 

500  5  40!  7  •.:•?   5500 

3  4 

13  26 

30 

3 

30 

20 

600  5  55  J  6  49 

5600 

2  49 

14  3 

31 

3 

30 

19 

700  6  10 

6  17 

5700 

2  35 

14  39 

32 

4 

30 

18 

800  6  24 

5  46 

5800 

2  21 

15  13 

33 

4 

29 

17 

900  6  38 

5  15 

5900 

2  8 

15  46 

34 

4 

29 

16 

1000  6  61 

4  46 

6000 

1  55 

16  18 

35 

4 

29 

16 

1100  7  4 

4  17 

6100 

1  42 

16  48 

36 

5 

28 

14 

1200 

7  15 

3  50 

6200 

1  31 

17  16 

37 

5 

28 

13 

1300 

7  27 

3  24 

6300 

1  19 

17  42 

38 

5 

27 

12 

1400 

7  37 

2  59 

6400 

1  9 

18  6 

39 

5 

27 

11 

1500 

7  47 

2  35 

6500 

0  59 

18  28 

40 

6 

26 

10 

1600 

7  55 

2  14 

6600 

0  50 

18  48 

41 

6 

26 

9 

1700 

8  3 

1  53 

6700 

0  42 

19  6 

42 

7 

25 

8 

1800 

8  10 

1  35 

6800 

0  34 

19  21 

43 

7 

25 

7 

1900 

8  16 

1  18 

6900 

0  28 

19  33 

44 

7 

24 

6 

1 

2000 

8  21 

1  3 

7000  i  0  22 

19  44 

45 

6 

23 

6 

2100 

8  25 

0  51 

7100  0  17 

19  52 

46 

8 

23 

4 

2200 

8  29 

0  40 

7200 

0  14 

19  57 

47 

9 

22 

3 

2300 

8  31 

0  32 

7300 

0  11 

20  0 

48 

9 

21 

2 

2400 

8  32 

0  25 

7400 

0  9 

20  1 

49 

10 

21 

1 

2500 

8  32 

0  21 

7500 

0  8 

19  59 

50 

10 

20 

0 

2600 

8  31 

0  19 

7600 

0  8 

19  55 

51 

10 

19 

99 

2700 

8  29 

0  20 

7700 

0  9 

19  48 

52 

11 

19 

98 

2800 

8  26 

0  23 

7800 

0  11 

19  40 

53 

11 

18 

97 

2900 

8  23 

0  28 

7900 

0  15 

19  29 

54 

12 

17 

96 

3000 

8  18 

0  36 

8000 

0  19 

19  17 

55 

12 

17 

95 

3100 

8  12 

0  47 

8100 

0  24 

19  2 

56 

13 

16 

94 

3200 

8  6 

0  59 

8200 

0  30 

18  45 

57 

13 

15 

93 

3300 

7  58 

1  14 

8300 

0  37 

18  27 

58 

13 

15 

92 

3400 

7  50 

1  32 

8400 

0  45 

18  6 

59 

14 

14 

91 

3500 

7  41 

1  52 

8500 

0  53 

17  45 

60 

14 

14 

90 

3600 

7  31 

2  14 

8600 

1  3 

17  21 

61 

15 

13 

89 

3700 

7  21 

2  38 

8700 

1  13 

16  56 

62 

15 

13 

88 

3800 

7  9 

3  4 

8800 

1  25 

16  30 

63 

15 

12 

87 

3900 

6  58 

3  32 

8900 

1  36 

16  3 

64 

15 

12 

86 

4000 

6  45 

4  2 

9000 

1  49 

15  34 

65 

16 

11 

85 

4100 

6  32 

4  34 

9100 

2  2 

15  5 

66 

16 

11 

84 

4200 

6  19 

5  7 

9200 

2  16 

14  34 

67 

16 

11 

83 

4300 

6  5 

5  41 

9300 

2  30 

14  3 

68 

16 

10 

82 

4400 

5  51 

6  17 

9400 

2  45 

13  31 

69 

17 

10 

81 

4500 

5  36 

6  54 

9500 

3  0 

12  58 

70 

17 

10 

80 

4600 

5  21 

7  32 

9600 

3  16 

12  25 

71 

17 

10 

79 

4700 

5  6 

8  11 

9700 

3  32 

11  52 

72 

17 

10 

78 

4800 

4  51 

8  50 

9800 

3  48 

11  18 

73 

17 

10 

77 

4900 

4  35 

9  30 

9900 

4  4 

10  44 

74 

17 

9 

76 

6000  4  20 

10  10 

10000 

4  20 

10  10 

75 

17 

9 

75 

104 


TABLE  XC. 


Mean  Right  Ascensions  and  Declinations  of  50  principal  Fixed 
Stars,  for  the  beginning  of  1 840. 


Stars'  Name. 

Mag 

flight  Ascen. 

AnnualVar. 

Declination. 

Ann.  Var. 

h     in       s 

8 

o    /       /> 

„ 

1       Algemb 

2.3 

0     5     0.31 

+    3.0775 

14  17  38.82  N 

-r  20.051 

2    0  Andromedae 

2 

1     0  46.7 

3.309 

34  46  17.2   N 

19.35 

3       Polaris 

2.3 

1     2  10.38 

16.1962 

88  27  21.96N 

19.339 

4      Achernar 

1 

1  31  44.88 

2.2351 

58     3     5.13  S 

—  18.473 

5    a  Arietis 

3 

1  58     9.94 

3.3457 

22  42  11.81  N 

-1-  17.456 

6    a  Ceti 

2.3 

2  53  55.34 

-}-    3.1257 

3  27  30.09  N 

+  14.561 

7    a  Persei 

2.3 

3  12  55.97 

4.2280 

49  17     8.74  N 

13.371 

8      Aldebaran 

1 

4  26  44.77 

3.4264 

16  10  56.82N 

7.949 

9       Capella 

1 

5     4  52.67 

4.4066 

45  49  42.81  N 

4.793 

10      Rig  el 

1 

5     6  51.09 

2.8783 

8  23  29.29S 

—   4.620 

11    /?Tauri 

2 

5  16  10.96 

+    3.7820 

28  27  58.20  N 

-t-    3.825  ' 

12    yOrionis 

2 

5  16  33.1 

3.210 

6  11  55.3    N 

+    3.82 

13    a  Columbae 

2 

5  33  51.52 

2  1688 

34     9  47.41  S 

—   2.291 

14    a  Orionis 

1 

5  46  30.71 

3.2430 

7  22  17.14N 

+    1.191 

15       Canopus 

1 

6  20  24.18 

1.3278 

52  36  38.42  S 

1.778 

16       Sirius               . 

1 

6  38     5.76 

+    2.6458 

16  30     4.79S 

+    4.449 

17       Castor 

3 

7  24  23.06 

3.8572 

32  13  58.89N 

—   7.206 

18      Procyon 

1.2 

7  30  55.53 

3.1448 

5  37  48.92N 

8.720 

19      PoZZwz 

2 

7  35  31.07 

3.6840 

28  24  25.57  N 

8.107 

20    aHydrae 

2 

9  19  43.57 

2.9500 

7  58     4.83  S 

+  15.341 

21      Regulus 

1 

9  59  50.93 

+    3.2220 

12  44  49.70N 

—  17.356 

22    a  Ursae  Majoris 

1.2 

10  53  47.98 

3.8077 

62  36  48.93N 

19.221 

23   /JLeonis 

2.3 

11  40  53.69 

3.0660 

15  28     1.16N 

19.985 

24   /?  Virginia 

3.4 

11  42  21.4 

3.124 

2  40     2  6    N 

19.98 

25   y  Ursae  Majoris 

2 

11  45  22.93 

3.1914 

54  35     467N 

20.014 

26aSCrucis 

2 

12  17  43.7 

+    3.258 

62  12  47.  9S 

+  19.99 

2T       Spica 

1 

13  16  46.36 

3.1502 

10  19  24.39  S 

18.945 

28    0Centauri 

2 

13  57  18.0 

3.491 

35  34  41.9    S 

17.499 

29    a  Draconis 

3.4 

14     0     2.8 

1.625 

65     8  32.1    N 

—  17.37 

30      Arcturus 

1 

14    8  21.96 

2.7335 

20     1     7.67  N 

18.956 

31  a  2  Centauri 

1 

14  28  47.84 

+    4.0086 

60  10     6.24S 

+  15.152 

32  a  2  Librae 

3 

14  42     2.44 

3.3088 

15  22  18.25S 

15.256 

33    @  Ursae  Minoris 

3 

14  51   14.66 

—   0.2787 

74  48  34.18N 

—  14.712 

34  y  2  Ursae  Minoris 

3.4 

15  21     1.3 

—   0.179 

72  24  14.1    N 

12.81 

35    a  Coronae  Borealis 

2 

15  27  54.87 

+    2.5277 

27  15  27.71  N 

12.361 

36    o  Serpentis 

2.3 

15  36  23.43 

+    2.9386 

6  56     2.80  N 

—  11.770 

37   /?  Scorpii 

2 

15  56     8.68 

3.4729 

19  21  38.82S 

+  10.330  ! 

38      An  tares 

1 

16  19  36.49 

3.6625 

26     4  13.13  S 

8.519 

«$9    a  Herculis 

3.4 

17     7  21.30 

2.7317 

14  34  41.43N 

—   4.576 

40    aOphiuchi 

2 

17  27  30.56 

2.7724 

12  40  58.65  N 

2.844 

41    (5  Ursae  Minoris 

3 

18  23  56.48 

—  19/J072 

86  35  28.89  N 

+    2.161 

42      Vega 

1 

18  SI  31.19 

+    20116 

38  38  16.85N 

2.742 

43      Altair 

1 

19  42  58.61 

2.9255 

8  27     0.21N 

8.701 

44  a  2  Capricorn! 

3 

20     9  10.34 

3.3323 

13     2     5.57  S 

—  10705 

45    a  Cygni 

1 

20  35  58.80 

2.0416 

44  42  41.38N 

+  12.614 

46    a  Aquarii 

3 

21  57  33.93 

+    3.0835 

1     5  38.00  S 

—  17.256 

47      Fomalhaut 

1 

22  48  47.67 

33114 

30  28     4.91  S 

19.092 

48   /JPegasi 

2 

22  56     1.1 

2.878 

27  13     1.7   N 

+  19.255 

49      Markab 

2 

22  56  47.75 

2.9771 

14  20  46.92N 

19  295 

50    a  Andromedae 

1 

24    0     7.72          3.0704 

28  12  27.06  N 

20.05R 

TABLE  XCI. 


105 


Constants  for  the  Aberration  and  Nutation  in  Right  Ascension 
and  Declination  of  the  Stars  in  the  preceding  Catalogue 


Aberration. 

Nutaticn. 

0 

M    ||          « 

N 

V 

M' 

6- 

N' 

«    °    ' 

»    °     ' 

s    °      ' 

s    o     ' 

1 

8  28  47 

0.1087 

7  27   12   0  9657 

6     8   24 

0.0300 

5  28  30 

0.8381 

2 

8   13  39 

0.1830 

6   19   12  ;  1.0740 

6  19   53 

0.0838 

5  10     8 

0.8496 

2 

8  13  51    1.6525 

5  16  57  |  1.3052 

8  16     7 

1.3427 

5  10  22 

0.8493 

4 

8     5  20    0.3801 

10  26  46;  1.2798 

4  10   12 

0.0775 

5     0  31 

0.8629 

5 

7  28  26 

0.1397 

702   0.8972 

6  11      1 

0.0695 

4  22  53 

0.8765 

6 

7  14   11 

0.1149 

8  23     S   0  8678 

6     1   26 

0.0322 

4    8  16 

0.9078 

7 

7     9  30 

0.3020      535 

1.0630 

6  18   13 

0.1849 

4     3  47 

0.9179 

8 

6  21  43 

0.1447 

7  23  12 

0.5760 

6     3  27 

0.0726 

3  17  54 

0.9502 

9 

6  12  51 

0.2875 

3  25  37 

0.9112 

6     5  4fi 

0.1830 

3  10  29 

0.9605 

10 

6   12  20 

0.1355 

9     3  42 

1.0300 

5  28  47 

1.9966 

3  10     4 

0.9608 

11 

6  10  13 

0.1873 

4  19  21 

03917 

6     2  52 

0.1008 

3     8   19 

0.9626 

12 

6  10     6 

0.1340 

8  26     4 

0.7851 

6     0  40 

0.0441 

3     8   14 

09626 

13 

665 

0.2145 

9     4  24 

1.2348 

5  26   IS    1.8750 

3     4  57 

0.9648 

14 

6     3  13 

0.1361 

8  28  23 

0.7521 

6     0   15   0.0481 

3     2  37 

0.9657 

15 

5  25  22 

0.3491 

8  25  53 

1.2960 

6     8  46 

1.6679 

2  26   15 

0.9657 

16 

5  21  21 

0.1501 

8  25  51 

1.1152 

6     1  51 

1.9658 

2  22  58 

0.9636 

17 

5  10  40 

0.2010 

1     2  17 

0.6620 

5  24     2 

0.1257 

2  14     6 

09535! 

18 

596 

0.1297 

9     6  54 

0.8071 

5  28  47 

0.0414 

2  12  47 

0.9513 

19 

582 

0.1829 

0  14  32 

0.6052 

5  24     2 

0.1114 

2   11  53 

0.9499 

20 

4  12  39 

0.1158 

8  17  31 

0.9967 

6     3  41 

0.0081 

1    18  37 

0.9007 

21 

4     2  22 

0.1162 

10     3  47 

0.8457 

5  23  47 

0.0480 

1     7  59 

0.8782 

22 

3  18     7 

0.4366 

0     3  28 

1.2394 

4  18  58 

0.2407 

0  21  57 

0.8520 

23 

3     5  21 

0,1117 

10     6  20 

0.9621 

5  20  56 

0.0344 

0     6  35 

0.8393 

24 

3     4  57 

0.0958 

9     6  51 

0.9075 

5  28  25 

0.0253 

065 

0.8390 

25 

348 

0.3229 

11   17  28 

1.2298 

4  21  46 

0.1465 

055 

0.8383 

26 

2  25  19 

0.4261 

685 

1.2585 

7  16     2 

0.2089 

11   24  14 

0.8390 

27 

2     9  22 

0.1066 

8     3  31 

08862 

6     5  51 

0.0154 

11     5     6 

0.8559 

28 

1  28  40 

0.1942 

6     7  12 

1.0176 

6  17  31 

U.1062 

10  23     8    0.8760 

29 

1  27  53 

0.4824  ' 

10  23  28 

1.2995 

3  25  50 

0.1090 

10  22   16    0.8777 

30 

1  25  46 

0.1336 

9  28  18 

1.0974 

5  18  49 

1.9937 

10  20     1    0.8822 

31 

20  32 

0.4123 

5     7  54 

1.1820 

6  29     6 

0.2460 

10  14  36  !  0.8937 

32 

17  26 

0.1273 

7  18  24!  0.6923 

6     6  29 

0.0593!  10   11   28  ;  0.9006 

33 

14  42 

0.6961 

10  15     5    1.3087 

2  26  45 

0.2235 

10     8  47   0.9066 

34 

7  20   0.6386 

10     7  33    1.3087 

2  27     7 

0.0960 

10     1  45   0.9225 

35 

5  45   0.1704 

9  22  28 

1.1785 

5  17  18 

T.9510 

10     0   18    0.9257 

36 

1     3  43   0.1237 

9     8  22 

0.9994 

5  27  30 

0.0068 

9  28  26    0.9298 

37 

0  28  58   0.1485 

744   0.6237 

6     5  20 

0.0795 

9  24  12  j  0.9386 

38 

0  23  24  0.1723 

5  27  59   0.5816 

6     5  49 

0.1029 

9    i9  21  i  0.9478 

39 

0  12  13   0.1451 

9     5  25 

1.0962 

5  27  45    1.9742 

9     9  58    0.9610 

40 

0     7  34  0.1427 

934 

1.0786 

5  28  48 

1.9803 

969 

0.9642 

41 

11  23  47   1.3571 

8  22  49 

1.2821 

11   19  31 

0.8257 

8  24  57 

0.9650 

42 

11  22  50   0.2393 

8  24  29 

1.2545 

6     5  31 

1.8436 

8  24  10   0.9644 

43 

11     6  15   0.1309 

8  22  59 

1.0237 

6     2  16 

1.998S 

8   10  21    0.9472. 

44 

11     0     2   0.1341 

9  29  33 

0.6961 

5  26  12 

0061)9 

8     4  55   0.9368 

45 

10  23  29   0.2668 

8     0  39 

1.2634 

6  28  32 

1.9042 

7  29     0   0.9242 

46 

10     2  57  !  0.1057 

9     2  31 

0.8988 

5  29  26 

0.0264 

7     8  37 

0.8794 

47 

9  19  26   0.1638 

11     7  34 

1.0271 

5  13     8 

0.0765 

6  23  30 

0.8540 

48 

9  17  29:0.1491 

7  17     0 

1.1171 

6  17     2 

0.0162 

6  21    13  i  0.8511 

49 

9  17  17  0.1120 

825 

1.0138 

6     8  23 

0.0157 

6  20  58:0.8508 

50 

9     0     6   0.1495 

7     6  42 

1.0785 

6  17  20 

0.0444 

6     0     8  i  0.8380 

106 


TABLE  XCII. 


Mean  Longitudes  and  Latitudes  of  some  of  the  principal  Fixed 
Stars  for  the  beginning  of  1840,  with  their  Annual  Variations 


Stars'  Name. 

Mag 

Longitude. 

Annual 
Var. 

Latitude. 

Annual 
Var. 

,            0      '          » 

// 

O      '           " 

/' 

a  Arietis 

3 

1     5  25  27.6 

50.277 

9  57  40.9  N 

+  0.161 

Aldebaran 

1 

2     7  33     5.9 

50.210 

5  28  38.0  - 

—  0.335 

Capella 

1 

2  19  37  17.8 

50.302 

22  51  44.4  N 

—  0.052 

Polaris 

2.3 

2  26  19  20.1 

47.959 

66    4  59.5  N 

+  0.552 

Sirius 

1 

3  11  52  32.9 

49.488 

39  34    4.3  S 

-1-  0.319 

Canopus 

1 

3  12  44  59.6 

49.366 

75  50  57.6  S 

+  0.459 

Pollux 

2 

3  21     0  22.0 

49.502 

6  40  20.2  N 

+  0.255 

Regulus 

1 

4  27  36  13.2 

49.946 

0  27  38.3  N 

-1-  0.220 

Spica 

1 

6  21  36  29.2 

50.085 

2     2  29.7  S 

+  0.171 

Arcturus 

1 

6  22     0     4.7 

50.711 

30  51   17.5  N 

+  0.214 

Antares 

1 

8     7  31  45.2 

50.120 

4  32  51.6  S 

4  0.424 

Altair 

1.2 

9  29  31     5.9 

50.795 

29  18  37.3  N 

+  0.080 

Fomalhaut 

1 

11     1  36  22.0 

50.595 

21     6  49.7  S 

+  0.213 

Achernar 

1 

11   13     2     5.3 

50.346 

17     6  17.3  S 

—  0.083! 

a  Pegasi 

2 

11  21   15  24.7 

50.112 

19  24  40.9  N 

+  0.098  ' 

TABLE  added  to  TABLE  XC. 

Mean  Right  Ascensions  and  Declinations  of  Polaris  and 
Minoris  for  1830,  1840,  1850,  and  1860. 


Ursat 


Stars. 

Years 

Right  Asc. 

Ann.  Var. 

Declination. 

Ann.  Var. 

o      •       >. 

,, 

o     /        •/ 

,, 

1830 

0  59  30.76 

+  15.478 

88  24    8.82 

+  19.371 

Polaris 

1840 
1850 

1     2  10.32 
1     5     0.29 

16.470 
17.567 

88  27  22.43 
88  30  35.40 

19.309 
19.240 

1880 

1     8     1.79 

18.784 

88  33  47.64 

19.163 

1830 

18  27    5.13 

—  19.167 

86  35     5.70 

+    2.363 

<J  Uisae  Minoris 

1840 

18  23  53.03 

19.241 

86  35  27.93 

2.085 

1850 

18  20  40.21 

19.305 

86  35  47.36 

1.805 

1860 

18  17  26.77 

19.360 

86  36    3.97 

1.623 

TABLE  XO.  (a). 

Mean  Places  of  50  Principal  Fixed  Stars. 
For  January  Od.,  1870. 


107 


Star's  Name. 

Mag. 

Eight  Aseen. 

Annual  Var 

Declination. 

Annual  Var. 

a  Andromedae  
y  Pegasi  (Algenib).  . 
Urs.  Min.  (Polaris 
Eridani  (  Achernar 
Arietis    

2 
3.2 
2 

I 
2 

h.  m.       ». 
0     1  40.238 
0     6  32.648 
1  11  16  990 
1  32  52.02h 
1  59  50  914 

8. 

+   3.0864 
3.0811 
20.1966 
2.2349 
3  3665 

N.  28°  22'  21".62 
N.  14  27  38  .40 
X.  88  36  58  .74 
S.  57  53  51  .51 
N.  22  50  46  .85 

+  19."899 
20  .027 
19  .091 
18  .419 
17  .224 

*Ceti  

2.3 

2  55  29.089 

+   3.1273 

N.  3  34  40  .00 

+  14  .344 

Persei  

2 

3  15     3.176 

4.2481 

1ST.  49  23  44  .88 

13  .169 

Tauri  (Aldebaran). 
Aungae  (Capella). 
$  Orionis  (Rigd).... 

/?  Tauri  

1 

1 
1 

2 

4  28  27.782 
5     7     5.338 
5     8  17.411 

5  18     4.487 

3.4353 
4.4217 
2.8799 

+   37873 

N.  16  14  44  .14 
N.  45  51  44  .60 
S.  8  21  15  .13 

N.  28  29  40  .42 

7  .622 
4  .154 
4  .464 

+   3  .446 

2 

5  25  21.975 

8.0641 

S.  0  23  52  63 

2  .980 

a  Columbas 

2 

5  34  56  721 

2.1778 

SL  34  8  40  .13 

2  .187 

a  Orionis  

Yar. 

5  48     8.026 

3.2462 

N.  7  22  48  64 

+   1  .035 

a  Argds((7a7zopws).  . 

a  Can  is  Maj.  (Sinus] 
aa  Geminor  (Castor), 
a  C&n\s}im.(Procyon] 
8  Geminor  (Pollux), 
a  Hydrje  

1 

1 
2.1 
1 
1.2 
2 

6  21     6.083 

6  39  25  283 
7  26  18.152 
7  32  29.688 
7  37  21.465 
9  21  11.889 

1.3303 

+   2.6452 
3.8417 
3.1446 
3.6812 
2.9485 

S.  52  37  32  .06 

&  16  32  24  .53 

N.  32  10  14  .97 
N.  5  33  22  .44 
N.  28  20  15  .56 
S.  8  5  47  .54 

—  1  .842 

—  6  .657 
7  .651 
8  .908 
8  .324 
15  .397 

a  Leon  is  (Regulus).'. 
a  Ursse  Majoris  
0  Leonis  

1.2 
2 
2 

10     1  26.776 
10  55  41.171 
11  42  25.582 

+   3.2023 
3.7653 
3.0648 

N.  12  36  5  .31 
N.  62  27  7  .09 
N.  15  17  55  .39 

—17  .423 
19  .360 
20  .099 

y  Ursae  Majoris  
TJ  Virgiuis 

2.3 
34 

11  46  58.914 
12  13  15  244 

3.1887 
30650 

N.  54  25  2  .81 
N.  0  3  21  .65 

20  .027 
20  .054 

a1  Crucis  

1 

12  19  2-^.701 

+   3  2650 

S.  62  22  38  .17 

—19  .932 

a  Virginia  (Spicei)  .  . 
£  Yirginis  .     .    . 

1 
34 

13  18  20.737 
13  28     4232 

8.1506 
3.0523 

S.  10  28  55  .45 
N.  0  4  11  .64 

18  .932 
18  .527 

a  Bootis  (Arctunts)  . 
a*  Centauri 

1 
1 

14     9  43.897 
14  30  48  325 

2.7338 
40367 

N.  19  51  37  .41 
S.  60  17  39  .13 

18  .903 
15  .022 

t  Bootis 

2  3 

14  39  18  503 

+   26194 

N.  27  37  24  .36 

—15  .395 

a*  Libras  

23 

14  43  41  335 

+   8.3058 

S.  15  29  59  .61 

15  .211 

&  Ursae  Minoris.  
0  Libras  ... 

2 
2 

14  51     6.857 
15  10     0758 

—  0.2489 
+   3  2188 

N.  74  41  11  .24 
S.  8  54  8  .16 

14  .757 
13  .562 

a  Coronae  Boreal  is  .  . 
a  Serpentis     

2 
2.3 

15  29  10.990 
15  37  51.859 

2.5377 

+   2.9492 

N.  27  9  13  .63 

N".  6  50  11  .26 

12  .335, 
—11  .598 

/J1  Scorpii  

2 

15  57  52.802 

34776 

S.  19  26  50  .27 

10  .206 

a  Scorpii  (Antares).  . 

1.2 
Var. 

16  21  26.333 
17     8  43.136 

3.6668 
2.7322 

S.  26  8  27  .62 
X.  14  32  26  .06 

8  .387 
4  .404 

a  Ophiuchi    

2 

17  28  53.949 

+   2.7808 

N.  12  39  24  .42 

—  2  .92J. 

J  Ursae  Minoris.  .... 
a  Lyrse  (Vega)  
a  Aquilce  (Altair).  .  . 

4.5 
1 
1.2 
3.4 

18  14  16.673 
18  32  32.155 
19  44  26.344 
20  10  50.295 

—19.3995 
+   2.0304 
2.9272 
3.3324 

2T.  86  36  21  .06 
N.  38  39  51  .25 
N.  8  31  37  .06 
S.  12  56  45  .08 

+   1  .260 
8  .126 
9  .210 
10  .840 

2.1 

20  36  59.960 

2.0430 

N.  44  49  Q  .98 

13  .690 

611  Cygni  

5  6 

21     1     3.949 

+   2  6737 

N".  38  ft  41  .36 

+  17  .495 

a  Aouarii      .  .     .    . 

3 

21  59     6275 

3  0823 

S.  0  57  1  81 

17    317 

a  ~Pis.&us(Fomalhaut) 
a  Pegasi  (Markab)  . 

1.2 
2 
3.4 

22  50  27.642 
22  58  17.136 
23  34     1.860 

3.3288 
2.9831 
2.4018 

S.  30  18  38  .37 
N.  14  80  23  .04 
N.  76  54  24  .86 

18  .966 
19  .312 
20  .077 

108 


TABLE  XCIII. 
Second  Differences. 


Huurn  &  Minutes. 

r 

2' 

3' 

4' 

6' 

6' 

r 

8' 

9-| 

10' 

11' 

A  m 

h  m 

rr 

" 

0  0 

12  0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0  10 

11  50 

0.4 

0.8 

1.2 

1.6 

2.0 

2.4 

2.9 

3.3 

3.7 

4.1 

4.5 

0  20 

11  40 

0.8 

1.6 

2.4 

3.2 

4.1 

4.9 

5.7 

6.5 

7.3 

81 

8.9 

0  30 

11  30 

1.2 

2.4 

3.6 

4.8 

6.0 

7.2 

8.4 

9.6 

10.8 

12.0 

13.2 

0  40 

11  20 

1.6 

3.1 

4.7 

6.3 

7.9 

9.4 

11.0 

12.6 

14.2 

15.7 

17.3 

0  50 

11  10 

1.9 

3.9 

5.8 

7.8 

9.7 

11.6 

13.6 

15.5 

17.4 

19.4 

21.4 

1  0 

11  0 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.0 

18.3 

20.6 

22.9 

25.2 

1  10 

10  50 

2.6 

5.3 

7.9 

10.5 

13.2 

15.8 

18.4 

21.1 

23.7 

26.3 

29.0 

1  20 

10  40 

3.0 

5.9 

8.9 

11.9 

14.8 

17.8 

20.7 

23.7 

26.7 

29.6 

32.6 

1  30 

10  30 

3.3 

6.6 

9.8 

13.1 

16.4 

19.7 

23.0 

26.3 

29.5 

32.8 

36.1 

1  40 

10  20 

3.6 

7.2 

10.8 

14.4 

17.9 

21.5 

25.1 

28.7 

32.3 

35.9 

39.5 

1  50 

10  10 

3.9 

7.8 

11.6 

15.5 

19.4 

23.3 

27.2 

31.0 

34.9 

38.8 

42.7 

2  0 

10  0 

4,2 

8.3 

12.5 

16.7 

20.8 

25.0 

29.2 

33.3 

37.5 

41.7 

45.8 

2  10 

9  50 

4.4 

8.9 

13.3 

17.8 

22.2 

26.6 

31.1 

35.5 

40.0 

44.4 

48.8 

2  20 

9  40 

4.7 

9.4 

14.1 

18.8 

23.5 

28.2 

32.9 

37.6 

42.3 

47.0 

51.7 

2  30 

9  30 

4.9 

9.9 

14.8 

19.8 

24.7 

29.7 

34.6 

39.6 

44.5 

49.5 

54.4 

2  40 

9  20 

5.2 

10.4 

15.6 

20.7 

25.9 

31.1 

36.3 

41.5 

46.7 

51.9 

57.0 

2  50 

9  10 

5.4 

10.8 

16.2 

21.6 

27.1 

32.5 

37.9 

43.3 

48.7 

54.1 

59.5 

3  0 

9  0 

5.6 

11.3 

16.9 

22.5 

28.1 

33.8 

39.4 

45.0 

50.6 

56.3 

61.9 

3  10 

8  50 

5.8 

11.7 

17.5 

23.3 

29.1 

35.0 

40.8 

46.6 

52.4 

58.3 

64.1 

3  20 

8  40 

6.0 

12.0 

18.1 

24.1 

30.1 

36.1 

42.1 

48.1 

54.2 

60.2 

66.2 

3  30 

8  30 

6.2 

12.4 

18.6 

24.8 

31.0 

37.2 

43.4 

49.6 

55.8 

62.0 

68.2 

3  40 

8  20 

6.4 

12.7 

19.1 

25.5 

31.8 

38.2 

44.6 

50.9 

57.3 

63.7 

70.0 

3  50 

8  10 

6.5 

13.0 

19.6 

26.1 

32.6 

39.1 

45.7 

52.2 

58.7 

65.2 

71.7 

4  0 

8  0 

6.7 

13.3 

20.0 

28.7 

33.3 

40.0 

46.7 

53.3 

60.0 

66.7 

73.3 

4  10 

7  50 

6.8 

13.6 

20.4 

27.2 

34.0 

40.8 

47.6 

54.4 

61.2 

68.0 

74.8 

4  20 

7  40 

6.9 

13.8 

20.8 

27.7 

34.6 

41.5 

48.4 

55.4 

62.3 

69.2 

76.1 

4  30 

7  30 

7.0 

14.1 

21.1 

28.1 

35.2 

42.2 

49.2 

56.2 

63.3 

70.3 

77.3 

4  40 

7  20 

7.1 

14.3 

21.4 

28.5 

35.6 

42.8 

49.9 

57.0 

64.2 

71.3 

78.4 

4  50 

7  10 

7.2 

14.4 

21.6 

28.9 

36.1 

43.3 

50.5 

57.7 

64.9 

72.2 

79.4 

5  0 

7  0 

7.3 

14.6 

21.9 

29.2 

36.5 

43.8 

51.0 

58.3 

65.6 

72.9 

80.2 

5  10 

6  50 

7.4 

14.7 

22.1 

29.4 

36.8 

44.1 

51.5 

58.8 

66.2 

73.6 

80.9 

5  20 

6  40 

7.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.9 

59.3 

66.7 

74.1 

81.5 

5  30 

6  30 

7.4 

14.9 

22.3 

29.8 

37.2 

44.7 

52.1 

59.6 

67.0 

74.5 

81.9 

5  40 

6  20 

7.5 

15.0 

22.4 

29.9 

37.4 

44.9 

52.3 

59.8  67.3 

74.8 

82.2 

5  50 

6  10 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0  67.4 

74.9 

82.4 

6  0 

6  0 

7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5 

60.0  67.5 

75.0 

82.5 

TABLE  XCIII. 
Second  Differences. 


109 


Hours  &Min.  10" 

20" 

30" 

40" 

50"i:  i" 

2" 

3" 

4" 

B"!6"| 

7" 

8" 

9" 

"  "' 

0  0  12  0  0.0 

0.0 

0.0 

0.0 

0.0  0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

iO  10  11  50  0.1 

0.1 

0.2 

0.3 

0.3 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

iO  20  11  40)0.1 

0.3 

0.4 

0.5 

0.7 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0  30  11  30  0.2  0.4 

0.6 

0.8 

1.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0  40  11  20  0.3  0.5 

0.8 

1.0 

1.3 

0.0 

0. 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0  50  11  10 

0.3 

0.6 

1.0 

1.3 

1.6 

0.0 

0. 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0 

11  0 

0.4 

0.8 

1.1 

1.5 

1.9 

0.0 

0. 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

10 

10  50 

0.4 

0.9  1.3 

1.8 

2.2 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4  1  0.4 

20 

10  40 

0.5 

1.0  1.5 

2.0 

2.5 

0.0 

0. 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

30 

10  30 

0.5 

1.1  1.6 

2.2 

2.7 

0.1 

0. 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

40 

10  20 

0.6 

1.2 

1.8 

2.4 

3.0 

0.1 

0. 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.5 

50 

10  10 

0.6 

1.3 

1.9 

2.6 

3.2 

0.1 

0. 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

2  0 

10  0 

0.7 

1.4 

2.1 

2.8 

3.5 

0.1 

0.1  0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.6 

2  10 

9  50 

0.7 

1.5 

2.2 

3.0 

3.7 

0.1 

0.1  0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

2  20 

9  40 

0.8 

1.6 

2.3 

3.1 

3.9 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

2  30 

9  30 

0.8 

1.6 

2.5 

3.3 

4.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

2  40 

9  20 

0.9  1.7 

2.6|3.5 

4.3 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

2  50 

9  10 

0.9 

1.8 

2.7  1  3.6  j  4.5 

0.1 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

0.8 

3  0 

9  0 

0.9 

1.9 

2.8 

3.8 

4.7 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

0.8 

3  10 

8  50 

1.0 

1.9 

2.9 

3.9 

4.9 

0.1 

0.2 

0.3 

0.4  0.5 

0.6 

0.7 

0.8 

0.9 

3  20 

8  40 

1.0  2.0 

3.0 

4.0 

5.0 

0.1 

0.2 

0.3 

0,4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  30 

8  30 

1.0  2.1 

3.1 

4.1 

5.2 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

3  40 

8  20 

1.1  2.1 

3.2 

4.2 

5.3 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

3  50 

8  10 

1.1 

2.2 

3.3 

4.3 

5.4 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

1.0 

4  0 

8  0 

1.1 

2.2 

3.3 

4.4  1  5.6 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.8 

0.9 

1.0 

4  10 

7  50 

l.l|2.3 

3.4 

4.5  5.7 

0.1 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  20 

7  40 

1.2 

2.3 

3.5 

4.6 

5.8 

0.1 

0.2 

0.3 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

4  30 

7  30 

1.2 

2.3 

3.5 

4.7 

5.9 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1. 

4  40 

7  20 

1.2 

2.4 

3.6 

4.8 

5.9 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1. 

4  50 

7  10 

1.2 

2.4 

3.6 

4.8 

6.0 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1. 

5  0 

7  0 

1.2 

2.4 

3.6 

4.9 

6.1 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1. 

5  10 

6  50 

1.2 

2.5 

3.7 

4.9 

6.1 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1. 

5  20 

6  40 

1.2 

2.5 

3.7 

4.9 

6.1 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1. 

5  30 

6  30 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1. 

5  40 

6  20 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1. 

5  50 

6  10 

1.2 

2.5 

3.7 

5.0 

6.2 

0.1 

0.2 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.1 

6  0 

6  0 

1.3 

2.6 

38 

5.0 

6.3i  0.1 

0.2 

0.4 

0.5 

0.6  |  0.7 

0.9  1  1.0  j  1.1 

110 


TABLE  XCIV. 
Third  Differences. 


Time  after 
noon  or 
midnight. 

10" 

20" 

30" 

40" 

50" 

1' 

2' 

3' 

4' 

5' 

Time  after 
noon  or 

midnight. 

Oh.Om. 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

12h.   Om. 

0    30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.5 

0.7 

0.9 

11     30 

1       0 

0. 

0.1 

0.2 

0.2 

0.3 

0.3 

0.6 

1.0 

1.3 

1.5 

11       0     . 

1     30 

0. 

0.1 

0.2 

0.3 

0.3 

0.4 

0.8 

1.2 

.6 

2.1 

10    30 

2      0 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

.9 

2.3 

10      0 

2     30 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

1.0 

1.4 

.9 

2.4 

9    30 

3      0 

0. 

0.2 

0.2 

0.3 

0.4 

0.5 

0.9 

1.4 

.9 

2.3 

9      0 

3    30 

0. 

0.1 

0.2 

0.3 

0.4 

0.4 

0.9 

1.3 

.7 

2.2 

8     30 

4      0 

0. 

0.1 

0.2 

0.2 

0.3 

0.4 

0.7 

1.1 

.5 

1.9 

8      0 

4    30 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.6 

0.9 

1.2 

1.5 

7    30 

5      0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.6 

08 

1.0 

7      0 

5    30 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

6     30 

6      0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

6      0 

TABLE  XCV. 
Fourth  Differences. 


Time  after 
noon  or 

midnight. 

10" 

20" 

30" 

40" 

50" 

1' 

2' 

3' 

Time  after 
noon  or 

midnight. 

h.    m. 

h.     m. 

0      0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

12       0 

0    30 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.4 

0.6 

11     30 

1      0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.8 

1.2 

11       0 

1     30 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

1.2 

1.7 

10    30 

2      0 

0.1 

0.2 

04 

0.5 

0.6 

0.7 

1.5 

2.2 

10      0 

2    30 

0.1 

0.3 

0.4 

0.6 

0.7 

0.9 

1.8 

2.7 

9    30 

3      0 

0.2 

0.3 

0.5 

0.7 

0.9 

1.0 

2.1 

3.1 

9      0 

3    30 

0.2 

0.4 

0.6 

0.8 

0.9 

1.1 

2.3 

3.4 

8    30 

4      0 

0.2 

0.4 

0.6 

0.8 

1.0 

1.2 

2.5 

3.7 

8      0 

4    30 

0.2 

0.4 

0.7 

0.9 

1.1 

1.3 

2.6 

3.9 

7    30 

5      0 

0.2 

0.5 

0.7 

0.9 

1.1 

1.4 

2.7 

4.1 

7      0 

5    30 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6    30 

6      0 

0.2 

0.5 

0.7 

0.9 

1.2 

1.4 

2.8 

4.2 

6      0 

TABLE  XCVI.     Logistical  Logarithms. 


Ill 


^ 

0 

1  I   2 

3 

4 

5    6 

7 

8 

b 

0 

60 

120 

180 
1.3010 

240 

300 

360 

420 

480 

540 

•o 

1.7782 

1.4771 

1.1761 

1.0792 

1.0000 

9331  !  8751 

8239 

1 

3.5563 

1.7710 

1.4735 

1.2986 

1.1743 

1.0777 

9988 

9320  8742 

8231 

2 

3.2553 

1.7639  1.4699 

1.2962 

1.1725 

1.0763 

99  3 

9310  8733 

8223 

3 

3.0792 

1.7570|  1.4664 

1.2939 

1.1707 

1.0749 

9964 

9300  8724 

8215 

4 

2.9542 

1.7501 

1.4629 

1.2915 

1.1689 

1.0734 

9952 

9289  8715 

8207 

5 

2.8573 

1.7434 

1.4594 

1.2891 

1.1671 

1.0720 

9940 

9279 

8706 

8199 

6 

2.7782 

1.7388 

1.4559 

1.2868 

1.1654 

1.0706 

9928 

9269 

8697 

8191 

7 

2.7112 

1.7302 

1.4525 

1.2845 

1.1636 

1.0692 

9916 

9259 

8688 

8183 

8 

2.6532 

1.7238 

1.4491 

1.2821 

1.1619 

1.0678 

9905 

9249  i  8679 

8175 

9 

2.6021 

1.7175 

1.4457 

1.2798 

1.1601 

1.0663 

9893 

9238  |  8670 

8167 

10 

2.5563 

1.7112 

1.4424 

1  2775 

1.1584 

1.0649 

9881 

9228  8661 

8159 

11 

2.5149 

1.7050 

1.4390 

1.2753 

1.1566 

1.0635 

9869 

9218 

8652 

8152 

12 

2.4771 

1.6990 

1.4357 

12730 

1.1549 

1.0621 

9858 

9208 

8643 

8144 

13 

2.4424 

1.6930 

1.4325 

1.2707 

1.1532 

1.0608 

9846 

9198 

8635 

8136 

14 

2.4102 

1.6871 

1.4292 

1.2685 

1.1515 

1.0594 

9334 

9188 

8626 

8128 

15 

2.3802 

1.6812 

1.4260 

1.2663 

1.1498 

1.0580 

9823 

9178 

8617 

8120 

16 

2.3522 

1.6755 

1.4228 

1.2640 

1.1481 

1.0566 

9811 

9168 

8608 

8112 

17 

2.3259 

1.6698 

1.4196 

1.2618 

1.1464 

1.0552 

9800 

9158 

8599 

8104 

18 

2.3010 

1.6642 

1.4165 

1.2596 

1.1447 

1.0539 

9788 

9148 

8591 

8097 

19 

2.2775 

1.6587 

1.4133 

1.2574 

1.1430  |  1.0525 

9777 

9138 

8582 

8089 

20 

2.2553 

1.6532 

1.4102 

1.2553 

1.1413 

1.0512 

9765 

9128 

8573 

8081 

21 

2.2341 

1.6478 

1.4071 

1.2531 

1.1397 

1.0498 

9754 

9119 

8565 

8073 

22 

22139 

1.6425 

1.4040 

1.2510 

1.1380 

1.0484 

9742 

9109 

8556 

8066 

23 

2.1946 

1.6372 

1.4010 

1.2488 

1.1363 

1.0471 

9731 

9099 

8547 

8058 

24 

2.1761 

1.6320 

1.3979 

1.2467 

1.1347 

1.0458 

9720 

9089 

8539 

8050 

25 

2.1  584 

1.6269 

1.3949 

1.2445 

1.1331 

1.0444 

9708 

9079 

8530 

8043 

26 

2.1413 

1.6218 

1.3919 

1.2424 

1.1314 

1.0431 

9697 

9070 

8522 

8035 

27 

2.1249 

1.6168 

1.3890 

1.2403 

1.  1298  i  1.0418 

9686 

9060 

8513 

8027 

28 

2.1091 

1.6118 

1.3860 

1.2382 

1.1282i  1.0404 

9675 

9050 

8504 

8020 

29 

2.0939 

1.6069 

1.3831 

1.2362 

1.1266;  1.0391 

9664 

9041 

8496 

8012 

30 

2.0792 

1.6021 

1.3802 

1.2341 

1.1249  1.0378 

9652 

9031 

8487 

8004 

31 

2.0649 

1.5973 

1.3773 

1.2320 

1.1233  1.0365 

9641 

9021 

8479 

7997 

32 

2.0512 

1.5925 

1.3745 

1.2300 

1.1217!  1.0352 

9630 

9012 

8470 

7989 

33 

2.0378 

1.5878 

1.3716 

1.2279 

1.1201  1.0339 

9619 

9002 

8462 

7981 

34 

2.0248 

1.5832 

1.3688 

1.2259 

1..1.86  1.0326 

9608 

8992 

8453 

7974 

35 

2.0122 

1.5786 

1.3660 

1.2239 

1.1170 

1.0313 

9597 

8983 

8445 

7966 

36 

2.0000 

1.5740 

1.3632 

1.2218 

1.1154 

1.0300 

9586 

8973 

8437 

7959 

37 

1.98S1 

1.5695 

1.3604 

1.2198 

1.1138 

1.0287 

9575 

8964 

8428 

7951 

38 

1.9765 

1.5651 

1.3576 

1.2178 

1.1123 

1.0274 

9564 

8954 

8420 

7944 

39 

1.9652 

1.5607 

1.3549 

1.2159 

1.1107 

1.0261 

9553 

8945 

8411 

7936 

40 

1.9542 

1.5563 

1.3522 

1.2139 

1.1091 

1.0248 

9542 

8935 

8403 

7929 

41 

1.9435 

1.5520 

1.3495 

1.2119 

1.1076 

1.0235 

9532 

8926 

8395 

7921 

42 

1.9331 

1.5477 

1.3468 

1.2099 

1.1061 

1.0223 

9521 

8917 

8386 

79i4 

43 

1.9228 

1.5435 

1.3441 

1.2080 

1.1045 

1.0210 

9510 

8907 

8378 

7906 

44 

1.9128 

1.5393 

1.3415 

1.2061 

1.1030 

1.0197 

9499 

8898 

8370  |  7899 

45 

1.9031 

1.5351 

1.3388 

1.2041 

1.1015 

!  1.0185 

9488 

8888 

8361  |  7891 

46 

1.8935 

1.5310 

1.3362 

1.2022 

1.0999 

1.0172 

9478 

8879 

8353  7884 

47 

1.8842 

1.5269 

1.3336 

1.2003 

1.0984 

1.0160 

9467 

8870 

8345 

7877 

48 

1.8751 

1.5229 

1.3310 

1.1984 

1.0969 

1.0147 

9456 

8861 

8337 

7869 

49 

1.8661 

1.5189 

1.3284 

1.1965 

1.0954 

1.0135   9446 

8851 

8328 

7862 

50 

1.8573 

1.5149 

1.3259 

1.1946 

1.0939 

1.01221  9435 

8842 

8320 

7855 

51 

1.8487 

1.5110 

1.3233 

1.1927 

1.0924 

1.0110!  9425 

8833 

8312 

7847 

52 

1.8403 

1.5071  1.3208 

1.1908 

1.0909i  1.0098|  9414 

8824  8304 

7840 

53 

1.8320  1.503*  1.3183 

1.1889 

1.089411.00851  9404 

881418296 

7832 

54 

1.8239  1.4994;  1.3158 

1.1871  1.0880  '  1.0073   9393 

8805  !  8288 

7825 

55 

1.8159 

1.4956 

1.3133  1.1852 

1.0865 

1.0061 

9383 

8796  8279 

7818 

56 

1.8081 

1.4918 

1.3108  1.1834 

1.0850 

1.0049 

9372 

8787  8271 

7811 

57 

1.8004  1.4881  j  1.3083  1.1816  1.0835 

1.0036 

9362 

8778  8263 

7803 

58 

1.7929  1.4844  13059  1.1797 

1.0821  1.0024 

9351 

8769  8255  7796 

59 

1.7855  1.4808 

1.3034 

1.1779 

1.0806  1.0012 

9341 

8760  8247  7789 

60 

1.7782  1.4771  1.301C 

1  1761 

1.0792  1.0000 

9331 

8751  8239  7782 

•— 


112       TABLE  XCVI.     Logistical  Logarithms. 


10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20  |  21 

II 

600 

ti60 

720 

780 

840 

900 

960 

1020 

1080 

1140 

1200 

1260 

~0 

7782 

7~368 

6990 

6642 

6320 

6021 

5740 

5477 

5229 

4994 

4771 

4559 

1 

7774 

7361 

6984 

6637 

6315 

6016 

5736 

5473 

5225 

4990 

4768 

4556 

2 

7767 

7354 

6978 

6631 

6310 

6011 

5731 

5469 

5221 

4986 

4764 

4552 

3 

7760 

7348 

6972 

6625 

6305 

6006 

5727 

5464 

5217 

4983 

4760 

4549 

4 

7753 

7341 

6966 

6620 

6300 

6001 

5722 

5460 

5213 

4979 

4757 

4546 

5 

7745 

7335 

6960 

6614 

6294 

5997 

5718 

5456 

5209 

4975 

4753 

4542 

6 

7738 

7328 

6954 

6609 

6289 

5992 

5713 

5452 

5205 

4971 

4750 

4539 

7 

7731 

7322 

6948 

6603 

6284 

5987 

5709 

5447 

5201 

4967 

4746 

4535 

8 

7724 

7315 

6942 

6598 

6279 

5982 

5704 

5443 

5197 

4964 

4742 

4532 

9 

7717 

7309 

6936 

6592 

6274 

5977 

5700 

5439 

5193 

4960 

4739 

4528 

10 

7710 

7002 

6930 

6587 

6269 

5973 

5695 

5435 

5189 

4956 

4735 

4525 

11  7703 

/296 

6924 

6581 

6264 

5968 

5691 

5430 

5185 

4952 

4732 

4522 

12 

7696 

7289 

6918 

6576 

6259 

5963 

5686 

5426 

5181 

4949 

4728 

4513 

13 

7688 

7283 

6912 

6570 

6254 

5958 

5682 

5422 

5177 

4945 

4724 

4515 

14 

768  i 

7276 

6906 

6565 

6248 

5954 

5677 

5418 

5173 

4941 

4721 

4511 

15 

7674 

7270 

6900 

6559 

6243 

5949 

5673 

5414 

5169 

4937 

4717 

4508 

16 

7667  1  '.'264 

6894 

6554 

6238 

5944 

5669 

5409 

5165 

4933 

4714 

4505 

17 

7660  1  7257 

6888 

6548 

6233 

5939 

5664 

5405 

5161 

4930 

4710 

4501 

18 

7653 

7251 

6882 

6543 

0228 

5935 

5660 

5401 

5157 

4926 

4707 

4498 

19 

7646 

7244 

6877 

6538 

b223 

5930 

5655 

5397 

5153 

4922 

4703 

4494 

20 

7639 

7238 

6871 

6532 

6218 

5925 

5651 

5393 

5149 

4918 

4699 

4491 

21 

7632 

7232 

6865 

6527 

6213 

5920 

5646 

5389 

5145 

4915 

4696 

4488 

22 

7625 

7225 

6859 

6521 

6208 

5916 

5642 

5384 

5141 

4911 

4692 

4484 

23 

7618 

7219 

6853 

6516 

6203 

5911 

5637 

5380 

5137 

4907 

4689 

4481 

24 

7611 

7212 

6847 

6510 

6198 

5906 

5633 

5376 

5133 

4903 

4685 

4477 

25 

7604 

7206 

6841 

6505 

6193 

5902 

5629 

5372 

5129 

4900 

4682 

4474 

26 

7597 

7200 

6836 

6500 

6188 

5897 

5624 

5368 

5125 

4896 

4678 

4471 

27 

7590 

7193 

6830 

6494 

6183 

5892 

5620 

5364 

5122  '  4892 

4675 

4467 

28 

7383 

7187 

6824 

6489 

6178 

5888 

5615 

5359 

5118  4889 

4671 

4464 

29  !  7577 

7181 

6818 

6484 

6173 

5883 

5611 

5355 

5114  4885 

4668 

4460 

30 

7570 

7r,s 

6812 

6478 

6168 

5878 

5607 

5351 

5110  4881 

4664 

4457 

31 

7563 

7168 

6807 

6473 

6163 

5874 

5602 

5347 

5106  4877 

4660 

4454 

32 

7556 

7162 

6801 

6467 

6158 

5869 

5598 

5343 

5102  4874 

4657 

4450 

33 

7549 

7156 

6795 

6462 

6153 

5864 

5594 

5339 

5098  '  4870 

4653 

4447 

34 

7542 

7149 

6789 

6457 

6148 

5860 

5589 

5335 

5094  ;  4866 

4650 

4444 

35 

7535 

7143 

6784 

6451 

6143 

5855 

5585 

5331 

5090  |  4863 

4646 

4440 

36 

7528 

7137 

6778 

6446 

6138 

5850 

5580 

5326 

5086  '  4859 

4643 

4437 

37 

7522 

7131 

6772 

6441 

6133 

5846 

5576 

5322 

5082  !  4855 

4639 

4434 

38 

7515 

7124 

6766 

6435 

6128 

5841 

5572 

5318 

5079  '  4852 

4636 

4430 

39 

7508 

7118 

6761 

6430 

6123 

5836 

5567 

5314 

5075  !  4848 

4632 

4427 

40 

7501 

7112 

6755 

6425 

6118 

5832 

5563 

5310 

5071 

4844 

4629 

4424 

41 

7494 

7106 

6749 

6420 

6113 

5827 

5559 

5306 

5067 

4841 

4625 

4420 

42 

7488 

7100 

6743 

6414 

6108 

5823 

5554 

5302 

5063 

4837 

4622 

4417 

43 

7481 

7093 

6738 

6409 

6103 

5818 

5550 

5298 

5059 

4833 

4618 

4414 

44 

7474 

7087 

6732 

6404 

6099 

5813 

5546 

5294 

5055 

4830 

4615 

4410 

45 

7467 

7081 

6726 

6398 

6094 

5809 

5541 

5290 

5051 

4826 

4611 

4407 

46 

7461 

7075 

6721 

6393 

6089 

5804 

5537 

5285 

5048 

4822 

4608 

4404 

47 

7454 

7069 

6715 

6388 

6084 

5800 

5533 

5281 

5044 

4819 

4604 

4400 

48 

7447 

7063 

6709 

6383 

6079 

5795 

5528 

5277 

5040 

4815 

4601 

4397 

49 

7441 

7057 

6704 

6377 

6074 

5790 

5524 

5273 

5036 

4811 

4597 

4394 

50 

7434 

7050 

6698 

6372 

6069 

5786 

5520 

5269 

5032 

4808 

4594 

4390 

51 

7427 

7044 

6692 

6367 

6064 

5781 

5516 

5265 

5028 

4804 

4590 

4387 

52 

7421 

7038 

6687 

6362 

6059 

5777 

5511 

5261 

5025  4800 

4587 

4384 

53 

7414 

7032 

6681 

6357 

6055 

5772 

5507 

5257 

5021 

4797 

4584 

4380 

54 

7407 

7026 

6676 

6351 

6050 

5768 

5503 

5253  5017 

4793 

4580 

4377 

55 

7401 

7020 

6670 

6346 

6045 

5763 

5498 

5249  ,5013 

4789  '  4577 

4374 

56 

7394 

7014 

6664 

6341 

6040 

5758 

5494 

5245 

5009 

4786  4573 

4370 

57 

7387 

7008 

6659 

6336 

6035 

5754 

5490 

5241 

5005 

47«?2  i  4570 

4367 

58 

7381 

7002 

6653 

6331 

6030  ,  5749 

5486 

5237 

5002 

4778  4506 

4364 

59 

7374 

6996  6648 

6325 

6025 

5745 

5481 

5233 

4998 

4775  !  4563 

4361 

60 

7368 

6990  i  bo42 

6320 

6021  16740 

5477 

5229 

4994 

4771  4559 

4357 

TABLE  XCVI.     Logistical  Logarithms. 


113 


—7  — 

22 

gj 

1380 

24 
T440 

25 

Isob 

26 
T560 

27 

28 
1680 

29 

3U 

31 

1860 

'32 

33 

1320 

1630 

1740 

1800 

1920 

IdbO 

"o 

4357  4164 

3979 

3802 

13632 

3468 

3310 

3158 

3010 

2868  2730 

2596 

1 

4354 

4161 

3976 

3799 

3629 

3465 

3307 

3155 

3008 

2866  ''•  2728 

2594 

2  4351  4158 

3973 

3796 

3626 

3463 

3305 

3153 

3005 

2863  2725 

2592 

3  4347  4155 

3970 

3793 

3623 

3460 

3302 

3150 

3003 

2861  2723 

2590 

4  434414152 

3967 

3791 

3621 

3457 

3300 

3148 

3001 

2859  2721 

2588 

5  4341  14149 

3964 

3738  3618 

3454 

3297 

3145 

2998 

2856  2719 

2585 

6  ;  4338  i  4145 

3961 

3785  '  3615 

3452 

3294 

3143 

2996 

2854  !27  16 

2583 

7  14334  4142 

3958 

3782  3612 

3449 

3292 

3140 

2993  !  ^52  2714 

2581 

8  j  4331  4139 

3955 

3779  |  36  10 

3446 

3289  j  3138 

2991 

28-'9  i2712 

2579 

9  4328  4136 

3952 

3776  3607 

3444 

3287 

3135 

2989 

2847  2710 

2577 

10  4325  4133 

3949 

3773  3604 

3441 

3284 

3133 

2986 

2845  2707 

2574 

11  |4321 

4130 

3946 

3770  3601 

3438 

3282 

3130 

2984 

2842  2/05 

2572 

12  4318 

4127 

3943 

3768  3598 

3436 

3279 

3128 

2981 

2840  2703 

2570 

13  4315.4124 

3940 

3765  3596 

3433 

3276 

3125 

2979 

2838 

2701 

2568 

14  14311  |4120 

3937 

3762  3593 

3431 

3274 

3123 

2977 

2835 

2698 

2566 

15  4308  1117 

3934 

3759  3590 

3428 

3271 

3120 

2974 

2833 

2696 

2564 

16  4305  4114 

3931 

3756 

3587 

3425 

3269 

3118 

2972 

2831 

2694 

2561 

17  4302;  41  11 

3928 

3753  3585 

3423 

3266 

3115 

2969 

2828 

2692 

2559 

18  4298:4108 

3925 

3750  3582 

3420 

3264 

3113 

2967 

2826 

2689 

2557 

19 

4295  |  4105 

3922 

3747  3579 

3417 

3261 

3110 

2965 

2824 

2687 

2555 

20 

4292  |  4102 

3919 

3745 

3576 

34  1£ 

3259 

3108 

2962 

2821 

2685 

2553 

21 

4289  I  4099 

3917 

3742 

3574 

3412 

3256 

3105 

2960 

2819 

2683 

2551 

22 

4285  4096 

3914 

3739 

3571 

3409 

3253 

3103 

2958 

2817 

2681 

2548 

23 

4282  i  4092 

3911 

3736 

3568 

3407 

3251 

3101 

2955 

2815 

2678 

2546 

24 

4279 

4039 

3908 

3733 

3565 

3404 

3248 

3098 

2953 

2812 

2676 

2544 

25 

4276 

4086 

3905 

3730 

3563 

3401 

3246 

3096 

2950 

2810 

2674 

2542 

26 

4273 

4083 

3902 

3727 

3560 

3399 

3243 

3093 

2948 

2808 

2672 

2540 

27 

4269 

4080 

3899 

3725 

3557 

3396 

3241 

3091 

2946 

2805 

2669 

2538 

28 

4266 

4077 

3896 

3722 

3555 

3393 

3238 

3088 

2943 

2803 

2667 

2535 

29 

4263 

4074 

3893 

3719 

3552 

3391 

3236 

3086 

2941 

2801 

2665 

2533 

30 

4260 

4071 

3890 

3716 

3549 

3388 

3233 

3083 

2939 

2798 

2663 

2531 

31 

4256 

4068 

3887 

3713 

3546 

3386 

3231 

3081 

2936 

2796 

2660 

2529 

32 

4253 

4065 

3884 

3710 

3544 

3383 

3228 

3078 

2934 

2794 

2658 

2527 

33 

4250 

4062 

3881 

3708 

3541 

3380 

3225 

3076 

2931 

2792 

2656 

2525 

34 

4247 

4059 

3878 

3705  3538 

3378 

3223 

3073 

2929 

2789 

2654 

2522 

35 

4244 

4055 

3875 

3702  i  3535 

3375 

3220 

3071 

2927 

2787 

2652 

2520 

36 

4240 

4052 

3872 

3699  3533 

3372 

3218 

3069 

2924 

2785 

2649 

2518 

37 

4237 

4049 

3869 

3696  !  3530 

3370 

3215 

3066 

2922 

2782 

2647 

2516 

38 

4234 

4046 

3866 

3693 

3527 

3367 

3213 

3064 

2920 

2780 

2645 

2514 

39 

4231 

4043 

3863 

3691 

3525 

3365 

3210 

3061 

2917 

2778 

2643 

2512 

40 

4228 

4040 

3860 

3688 

3522 

3362 

3208 

3059 

2915 

2775 

2640 

2510 

41 

4224 

4037 

3857  3685 

3519 

3359 

3205 

3056 

2912 

2773 

2638 

2507 

42 

4221 

4034 

38551  3682  i  3516 

3357 

3203 

3054 

2910 

2771 

2636 

2505 

43 

4218 

4031 

3852  i  3679 

3514 

3354 

3200 

3052 

2908 

2769 

2634 

2503 

44 

4215 

4028 

3849  !  3677 

3511 

3351 

3198 

3049 

2905 

2766 

2632 

2501 

45 

4212 

4025 

3846 

3674 

3508 

3349 

3195 

3047 

2903 

2764 

2629 

2499 

46 

4209 

4022 

3843 

3671 

3506 

3346 

3193 

3044 

2901 

2762 

2627 

2497 

47 

4205 

4019 

3840 

3668  j  3503 

3344 

3190 

3042 

2898 

2760 

2625 

2494 

48 

4202 

4016 

3837 

3665 

3500 

3341 

3188 

3039 

2896 

2757 

2623 

2492 

49 

4199 

4013 

3834 

3663 

3497 

3338 

3185 

3037 

2894 

2755 

2621 

2490 

50 

4196 

4010 

3831 

3660 

3495 

3336 

3183 

3034 

2891 

2753 

2618 

2488 

51 

4193 

4007 

3828  j  3657 

3492 

3333 

3180 

3032 

2889 

2750 

2616 

24%  6 

52 

4189 

4004 

3825  3654 

3489 

3331 

3178 

3030 

2887 

2748 

2614 

2484 

53 

4186 

4001  i  3322  3651 

3487 

3328 

3175 

3027 

2884 

2746 

2612 

2482 

54 

4183  !  3998  3820  3649  3484 

3325 

3173 

3025 

2882 

2744 

2610 

2480 

55 

4180  J3995  3317:3646 

3481 

3323  3170 

3022  2880  2741  1  2607 

2477 

56 

4177 

3991  '3814 

3643 

3479 

3320 

3168 

3020  '  2877 

2739 

2605 

2475 

57 

4174 

3988  3811 

3640  '  3476 

3318 

31651301812875 

2737 

2603 

2473 

58 

4171 

3985 

3308 

3637  3473 

3315 

3163 

30  IP-  ,2873 

2735  2601 

2471 

59 

4167 

3982 

3805 

3635  3471 

3313 

3160 

301312870 

2732 

2599 

2469 

60  4164 
i 

3979 

3802 

3632  3468 

3310 

3158  '3010  12868 

2730 

2596  i  2467 

114       TABLE  XCVJ.     Logistical  Logarithms. 


' 

34 

35 

36   37 

38 

39 

40 

41 

42 

43 

44 

45 

" 

2040 

2100 

2160;  2220 

2280 

2340 

2400 

2460 

"2520 

2580 

2040 

2700 

~7> 

2467 

2341 

2218  2099 

1984 

1871 

1761 

1654 

1549 

1447 

1347 

1249 

i 

2465 

2339 

2216  2098 

1982 

1869 

1759 

1652 

1547 

1445 

1345 

1248 

2 

2462 

2337 

2214  2096 

1980 

1867 

1757 

1650 

1546 

1443 

1344 

1246 

3 

2460 

2335 

2212  2094 

1978 

1865 

1755 

1648 

1544 

1442 

1342 

1245 

4 

2458 

2333 

2210  2092 

1976 

1863 

1754 

1647 

1542 

1440 

1340 

1243 

5 

2456 

2331 

2208 

2090 

1974 

1862 

17ft2  1645 

1540 

1438 

1339 

1241 

e 

2454 

2328 

2206 

2088 

1972 

1860 

1750  1643 

1539 

1437  1337 

1240 

7 

2452 

2326 

2204  2086 

1970 

1858 

1748  1641 

1537 

1435 

1335 

1238 

8 

2450 

2324 

2202  j  2084 

1968 

1856 

1746  i  1640 

1535 

1433 

1334 

1237 

9 

2448 

2322 

2200  i  2082 

1967 

1854 

1745  i  1638  j  1534 

1438  1  1332 

1235 

10 

2445 

2320 

2198 

2080 

1965 

1852 

1743 

1036 

1532 

1430 

1331 

1233 

11 

2443 

2318 

2196 

2078 

1963 

1850 

1741 

1634 

1530 

1428 

1329 

1232 

12 

2441 

2316 

2194 

2076 

1961 

1849 

1739 

1633 

1528 

1427 

1327 

1230 

13 

2439 

2314 

2192 

2074 

1959 

1847 

1737 

1631 

1527 

1425 

1326 

1229 

14 

2437 

2312 

2190 

2072 

1957 

1845 

1736 

1629 

1525 

1423 

1324 

1227 

16 

2-135 

2310 

2188 

2070 

1955 

1843 

1734 

1627 

1523 

1422 

1322 

1225 

16 

2433 

2308 

2186 

2068 

1953 

1841 

1732 

1626 

1522 

1420 

1321 

1224 

17 

2431 

2306 

2184 

2066 

1951 

1839 

1730 

1624 

1520 

1418 

1319 

1222 

18 

2429 

2304 

2182 

2064 

1950 

1838 

1728 

1622 

1518 

1417 

1317 

1221 

.19 

2426 

2302 

2180 

2062 

1948 

1836 

1727 

1620 

151b 

1415 

1316 

1219 

'20 

2424 

2300 

2178 

2061 

1946 

1834 

1725 

1619 

1515 

1413 

1314 

1217 

'21 

2422 

2298 

2176 

2059 

1944 

1832 

1723 

1617 

1513 

1412 

1313 

1216 

22 

2420 

2296 

2174 

2057 

1942 

1830 

1721 

1615 

1511 

1410 

1311 

1214 

23 

2418 

2294 

2172 

2055 

1940 

1828 

1719 

1613 

1510 

1408 

1309 

1213 

24 

2416 

2291 

2170 

2053 

1938 

1827 

1718 

1612 

1508 

1407 

1308 

1211 

25 

2414 

2289 

2169 

2051 

1936 

1825 

1716 

1610 

1506 

1405 

1306 

1209 

20 

2412 

2287 

2167 

2049 

1934 

1823 

1714 

1608 

1504 

1403 

1304 

1208 

27 

2410 

2285 

2165 

2047 

1933 

1821 

1712 

1606 

1503 

1402 

1303 

1206  ' 

28 

2408 

2283 

2163 

2045 

1931 

1819 

1711 

1605 

1501 

1400 

1301 

1205 

29 

2405 

2281 

2161 

2043 

1929 

1817 

1709 

1603 

1499 

1398 

1300 

1203 

30 

2403 

2279 

2159 

2041 

1927 

1816 

1707 

1601 

1498 

139? 

1298 

1201 

31 

2401 

2277 

2157 

2039 

1925 

1814 

1705 

1599 

1496 

1395 

1296 

1200 

32 

2399 

2275 

2155 

2037 

1923 

1812 

1703 

1598 

1494 

1393 

1295 

1198 

33 

2397 

2273 

2153 

2035 

1921 

1810 

1702 

1596 

1493 

1392 

1293 

1197 

34 

2395 

2271 

2151 

2033 

1919 

1808 

1700 

1594 

1491 

1390 

1291 

1195 

35 

2393 

2269 

2149 

2032 

1918 

1806 

1698 

1592 

1489 

1388 

1290 

1193 

36 

2391 

2267 

2147 

2030 

1916 

1805 

1696 

1591 

1487 

1387 

1288 

1192 

37 

2389 

2265 

2145 

2028 

1914 

1803 

1694 

1589 

1486 

1385 

1287 

1190 

38 

2387 

2263 

2143 

2026 

1912 

1801 

16<J3 

1587 

1484 

1383 

1285 

1189 

39 

2384 

2261 

2141 

2024 

1910 

1799 

1691 

1585 

1482 

1382 

1283 

1187 

40 

2382 

2259  -2139 

2022 

1908 

1797 

1089 

1584 

1481 

1380 

1282 

1186 

41 

2380 

2257 

2137 

2020 

1906 

1795 

1687 

1582 

1479 

1378 

1280 

1184 

42 

2378 

2255 

2135 

2018 

1904 

1794 

1686 

1580 

1477 

1377 

1278 

1182 

43 

2376 

2253 

2133 

2016 

1903 

1792 

1684 

1578 

1476 

1375 

1277 

1381 

44 

2374 

2251 

2131 

2014 

1901 

1790 

1082 

1577 

1474 

1373 

1275 

1179 

45 

2372 

2249 

2129 

2012 

1899 

1788 

1680 

1575 

1472 

1372 

1274 

1178 

46 

2370 

2247 

2127 

2010 

1897 

1786 

1678 

1573 

1470 

1370 

1272 

1176 

47 

2368 

2245 

2125 

2009 

1895 

1785 

1677 

1571 

1469 

1368 

1270 

1174 

48 

2366 

2243 

2123 

2007 

1893 

1783 

1675 

1570 

1467 

1307 

1269 

1173 

49 

2364 

2241 

2121 

2005 

1891 

1781 

1673 

1568 

1465 

1365 

1267 

1171 

60 

2362 

2239 

2119 

2003 

1889 

1779 

1671 

1566 

1464 

1363 

1266 

1170 

51 

2359 

2237 

2117 

2001 

1888 

1777 

1670 

1565 

1462 

1362 

1264 

1168 

52 

2357 

2235 

2115 

1999 

1886 

1775 

1668 

1563 

1460 

1360 

1262 

1107 

53 

2355 

2233 

2113 

1997 

1884 

1774 

1666 

1561 

1459 

1359 

1261 

1165 

54 

2353 

2231 

2111 

1995 

1882 

1772 

1664 

1559 

1457 

1357  1259'1163 

55 

2351 

2229 

2109 

1993 

1880 

1770 

1663 

1558  1  1455 

1355 

1257 

1162 

56 

2349 

2227 

2107 

1991 

1878 

1768 

1661 

1556  1454 

1354 

1256 

1100 

57 

2347 

2225  2105 

1989 

1876 

1766 

1659 

1554 

1452 

1352 

1254 

1159 

.58 

2345 

2223  i  2103 

1987 

1875 

1765 

1657 

1552 

1450 

1350 

1253 

1157 

59 

2343 

2220 

2101 

1986 

1873 

1763 

1655 

1551 

1449 

1349 

1251 

1156 

60 

2341 

2218 

2099  1984 

1871 

1761 

1654 

1549 

1447  1347  1249 

1154 

TABLE  XCVI. 


Logistical  Loganihins. 


115 


,  ' 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

?/ 

2760 

2820 

2880 

2940 

3000 

3060 

3120 

3180 

3240 

3300 

3360  j  3420 

3480 

3540 

0 

1154 

1061 

0969 

4)880  I  0792 

0706 

0621 

0539 

0458 

0378 

0300 

0223 

0147 

0073 

!   1 

1152 

1059 

0968 

0878 

0790 

0704 

0620 

0537 

0456 

0377 

0298 

0221 

0146 

0072 

2 

1151 

1057 

0966 

0877 

0789 

0703 

0619 

0536 

0455 

0375 

0297 

0220 

0145 

0071 

3 

1149 

1056 

0965 

0875 

0787 

0702 

0617 

0535 

0454 

0374 

0296 

0219 

0143 

0069 

4 

1148 

1054 

0963 

0874 

0786 

0700 

0616 

0533 

0452 

0373 

0294 

0218 

0142 

0068 

5 

1146 

1053 

0962 

0872 

0785 

0699 

0615 

0532 

0451 

0371 

0293 

0216 

0141 

0067 

6 

1145 

1051 

0960 

0871 

0783 

0697 

0613 

0531 

0450 

0370 

0292 

0215 

0140 

0066 

7 

1143 

1050 

0859 

0869 

0782 

0696 

0612 

0529 

0448 

0369 

0291 

0214 

0139 

0064 

8 

1141 

1048 

0957 

0868 

0780 

0694 

0610 

0528 

0447 

0367 

0289 

0213 

0137 

0063 

i  9)1140 

1047 

0956 

0866 

0779 

0693 

0609 

0526 

0446 

0366 

0288 

0211 

0136 

0062 

;  10  |1138 

1045 

0954 

0865 

0777 

0692 

0608 

0525 

0444 

0365 

0287 

0210 

0135 

0061 

11 

1137 

1044 

0953 

0863 

0776 

0690 

0606 

0524 

0443 

0363 

0285 

0209 

0134 

0060 

12 

1135 

1042 

09-51 

OS62 

0774 

0689 

0605 

0522 

0442 

0362 

0284 

0208 

0132 

0058 

13 

1134 

1041 

0950 

0860 

0773 

0687 

0603 

0521 

0440 

0361 

0283 

0206 

0131 

0057 

14 

1132 

1039 

0948 

0859 

0772 

0686 

0602 

0520 

0439 

0359 

0282 

0205 

0130 

0056 

15 

1130 

1037 

0947 

0857 

0770 

0685 

0601 

0518 

0438 

0358 

0280 

0204 

0129 

0055 

16 

1129 

1036 

0945 

0856 

0769 

0683 

0599 

0517 

0436 

0357 

0279 

0202 

0127 

0053 

17 

1127 

1034 

0944 

0855 

0767 

0682 

0598 

0516 

0435 

0356 

0278 

0201 

0126 

0052 

Us 

1126 

1033 

0942 

0853  |  0766 

0680 

0596 

0514 

0434 

0354 

0276 

0200 

0125 

0051 

19 

1124 

1031 

0941 

0852  |  0764 

0679 

0595 

0513 

0432 

0353 

0275 

0199 

0124 

0050 

20 

1123 

1030 

0939 

0850 

0763 

0678 

0594 

0512 

0431 

0352 

0274 

0197 

0122 

0049 

21 

1121 

1028 

0938 

0849 

0762 

0676 

0592 

0510 

0430 

0350 

0273 

0196 

0121 

0047 

22 

1119 

1027 

0936 

0847 

0760 

0675 

0591 

0509 

0428 

0349 

0271 

0195 

0120 

0046 

23 

1118 

1025 

0935 

0846 

0759 

0673 

0590 

0507 

0427 

0348 

0270 

0194 

0119 

0045 

24 

1116 

1024 

0933 

0844  |  0757 

0672 

0588 

0506 

0426 

0346 

0269 

0192 

0117 

0044 

25 

1115 

1022 

0932 

0843 

0756 

0670 

0587 

0505 

0424 

0345 

0267 

0191 

0116 

0042 

26 

1113 

1021 

0930 

0841 

0754 

0669 

0585 

0503 

0423 

0344 

0266 

0190 

0115 

0041 

27 

1112 

1019 

0929 

0840 

0753 

0668 

0584 

0502 

0422 

0342 

0265 

0189 

0114 

0040 

jfi 

1110 

1018 

0927 

0838 

0751 

0666 

0583 

0501 

0420 

0341 

0264 

0187 

0112 

0039 

29 

1109 

1016 

0926 

0837 

0750 

0665 

0581 

0499 

0419 

0340 

0262 

0186 

0111 

0038 

30 

1107 

1015 

0924 

0835 

0749 

0663 

0580 

0498 

0418 

0339 

0261 

0185 

0110 

0036 

31 

1105 

1013 

0923 

0834 

0747 

0662 

0579 

0497 

0416 

0337 

0260 

0184 

0109 

0035 

32 

1104 

1012 

0921 

0833 

0746 

0661 

0577 

0495 

0415 

0336 

0258 

0182 

0107 

0034 

33 

1102 

1010 

0920 

0831 

0744 

0659 

0576 

0494 

0414 

0335 

0257 

0181 

0106 

0033 

34 

1101 

1008 

0918 

0830 

0743 

0658 

0574 

0493 

0412 

0333 

0256 

0180 

0105 

0031 

35 

1099 

1007 

0917 

0828 

0741 

0656 

0573 

0491 

0411 

0332 

0255 

0179 

0104 

0030 

36 

1098 

1005 

0915 

0827 

0740 

0655 

0572 

0490 

0410 

0331 

0253 

0177 

0103 

0029 

|37 

1096 

1004 

0914 

0825 

0739 

0654 

0570 

0489 

0408 

0329 

0252 

0176 

0101 

0028 

38 

1095 

1002 

0912 

0824  0737 

0652 

0569 

0487 

0407 

0328 

0251 

0175 

0100 

0027 

39 

1093 

1001 

0911 

0822 

0736 

0651 

0568 

0486 

0406 

0327 

0250 

0174 

0099 

0025 

40 

1091 

0999 

0909 

0821 

0734 

0649  0566 

0484 

0404 

0326 

0248 

0172 

0098 

0024 

41 

1090 

0998 

0908 

0819  0733 

0648  0565 

0483 

0403 

0324 

0247 

0171 

0096 

0023 

42 

1088 

0996 

0906 

0818  0731 

0647  |  0563 

0482 

0402 

0323 

0246 

0170 

0095 

0022 

43 

1087 

0995 

0905 

0816  0730 

0645  0562 

0480 

0400 

0322 

0244 

0169 

0094 

0021 

44 

1085 

0993 

0903 

0815 

0729 

0644  0561 

0479 

0399 

0320 

0243 

0167 

0093 

0019 

45 

1084 

0992 

0902 

0814 

0727 

0642  0559 

0478 

0398 

0319 

0242 

0166 

0091 

0018 

46 

1082 

0990 

0900 

0312  0726 

0641  0558 

0476 

0396 

0318 

0241 

0165 

0090 

0017 

47 

1081 

0989 

0899 

0811  0724 

0640  0557 

0475 

0395 

0316 

0239  0163 

0089 

0016 

48 

1079 

0987 

0897 

0809  0723 

0638  ,  0555 

0474 

0394 

0315 

0238 

0162 

0088 

0015 

49 

1078 

0986 

0896 

0808  0721 

0637  0554 

0472 

0392 

0314 

0237 

0161 

0087 

0013 

50 

1076 

0984 

0894 

0806  0720 

0635  ;  0552 

0471 

0391 

0313 

0235 

0160 

0085 

0012 

01  1074 

0983 

0893 

0805  0719 

0634  0551 

0470 

0390 

0311 

0234  0158 

0084 

0011 

52  i  1073 

0981 

0891 

0803  0717 

0633 

0550 

0468 

0388 

0310 

0233  0157 

0083 

0010 

r)3  1071 

0980 

0890 

0802  0716 

0631 

0548 

0467 

0387 

0309 

0232 

0156 

0082 

0008 

54 

1070 

0978 

0888 

0801  !  0714 

0630 

0547 

0466 

0386 

0307 

0230 

0155 

008010007 

r)5 

1068 

0977  ,  0887 

0799  |  0713  •  0628 

0546 

0464 

0384  0306 

0229 

0153 

0079 

0006 

J56 

1067 

0975 

0885;  0798  0711  0627 

0544 

0463 

0383  !  0305 

0268 

0152 

0078 

0005 

57 

1065 

0974 

0884  !  0796 

0710 

0626 

0543 

0462 

0382 

0304  0227 

0151 

0077 

0004 

58 

1064 

0972 

0883 

0795 

0709 

0624  0541 

0460 

0381 

0302  0225 

0150 

0075 

0002 

59 

1062 

0971 

0881 

0793 

0707 

0623 

0540 

0459 

0379 

0301 

0224 

0148 

0074  0001 

60 

1061 

0969 

0880 

0792 

0706 

Of21 

0539 

0458  0378 

0300 

0223 

0147  0073,0000 

PLATE  I. — EQUATORIAL  TELESCOPE  OF  THE  OBSERVATORY  OF  HARVARD  COLLEGE. 


PLATE  II.— TOTAL  ECLIPSE  OF  THE  SUN,  OF  JULT  18,  1860,  AS  OBSERVED  BY 
DB.  FEILITZSCH,  AT  CASTELLOX  DE  LA  PL  AX  A. 


PLATE  III. — DOXATI'S  COJIET. 


PLATB  IV. — CLUSTESS  AND 


PLATE  V. — 


Q 


Fia.  t. 


FIG.  19. 


FIG.  23. 


FIG.  77. 


FIG.  66. 


FIG.  32. 


FIG.  98. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

»ook  is  due  on  the  last  d    e  ;tamp  v,  or 

on  the  date  to  wuic^  xonew... 
Renewed  books  are  subject  to  immediate  recall. 

MAY  1  2  1968  4 


RECEIVE 


12  '68  -6 


IWL 


LOAN 


— 


LD  21A-60?rc-10,'65 
(F7763slO)476B 


General  Library 

University  of  California 

Berkeley 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


